# A Quantum Mermin-Wagner Theorem for a Generalized Hubbard Model

Authors: Mark Kelbert; Yurii Suhov
Journal: Advances in Mathematical Physics
 (2013)

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## Abstract

This paper is the second in a series of papers considering symmetry properties of bosonic quantum systems over 2D graphs, with continuous spins, in the spirit of the Mermin-Wagner theorem. In the model considered here the phase space of a single spin isℋ1=L2(M), where M is a d-dimensional unit torus M=ℝd/ℤd with a flat metric. The phase space of k spins is ℋk=L2sym(Mk), the subspace of L2(Mk) formed by functions symmetric under the permutations of the arguments. The Fock space H=⊕k=0,1,…ℋk yields the phase space of a system of a varying (but finite) number of particles. We associate a space H≃H(i) with each vertex i∈Γ of a graph (Γ,ℰ) satisfying a special bidimensionality property. (Physically, vertex i represents a heavy “atom” or “ion” that does not move but attracts a number of “light” particles.) The kinetic energy part of the Hamiltonian includes (i) -Δ/2, the minus a half of the Laplace operator on M, responsible for the motion of a particle while “trapped” by a given atom, and (ii) an integral term describing possible “jumps” where a particle may join another atom. The potential part is an operator of multiplication by a function (the potential energy of a classical configuration) which is a sum of (a) one-body potentials U(1)(x), x∈M, describing a field generated by a heavy atom, (b) two-body potentials U(2)(x,y), x,y∈M, showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials V(x,y), x,y∈M, scaled along the graph distance d(i,j) between vertices i,j∈Γ, which gives the interaction between particles belonging to different atoms. The system under consideration can be considered as a generalized (bosonic) Hubbard model. We assume that a connected Lie group G acts on M, represented by a Euclidean space or torus of dimension d'≤d, preserving the metric and the volume in M. Furthermore, we suppose that the potentials U(1), U(2), and V are G-invariant. The result of the paper is that any (appropriately defined) Gibbs states generated by the above Hamiltonian is G-invariant, provided that the thermodynamic variables (the fugacity z and the inverse temperature β) satisfy a certain restriction. The definition of a Gibbs state (and its analysis) is based on the Feynman-Kac representation for the density matrices.


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## Body

## 1. Introduction

### 1.1. Basic Facts on Bi-Dimensional Graphs

As in [1], we suppose that the graph (Γ,ℰ) has been given, with the set of vertices Γ and the set of edges ℰ. The graph has the property that whenever edge (j′,j′′)∈ℰ, the reversed edge (j′′,j′) belongs to ℰ as well. Furthermore, graph (Γ,ℰ) is without multiple edges and has a bounded degree; that is, the number of edges (j,j′) with a fixed initial or terminal vertex is uniformly bounded: (1)sup[max(♯{j′∈Γ:(j,j′)∈ℰ},supamx5555♯{j′∈Γ:(j′,j)∈ℰ}):j∈Γ]<∞.
The bi-dimensionality property is expressed in the bound
(2)0<sup[1n♯Σ(j,n):j∈Γ,n=1,2,…]<∞,
where Σ(j,n) stands for the set of vertices in Γ at the graph distance n from j∈Γ (a sphere of radius n around j):
(3)Σ(j,n)={j′∈Γ:d(j,j′)=n}.
(The graph distance d(j,j′)=dΓ,ℰ(j,j′) between j,j′∈Γ is determined as the minimal length of a path on (Γ,ℰ) joining j and j′.) This implies that for any o∈Γ the cardinality ♯Λ(o,n) of the ball
(4)Λ(o,n)={j′∈Γ:d(o,j′)≤n}
grows at most quadratically with n.A justification for putting a quantum system on a graph can be that graph-like structures become increasingly popular in rigorous Statistical Mechanics, for example, in quantum gravity. Namely, see [2–4]. On the other hand, a number of properties of Gibbs ensembles do not depend upon “regularity” of an underlying spatial geometry.


### 1.2. A Bosonic Model in the Fock Space

With each vertexi∈Γ we associate a copy of a compact manifold M which we take in this paper to be a unit d-dimensional torus ℝd/ℤd with a flat metric ρ and the volume v. We also associate with i∈Γ a bosonic Fock-Hilbert space H(i)≃H. Here H=⊕k=0,1,…ℋk where ℋk=L2sym(Mk) is the subspace in L2(Mk) formed by functions symmetric under a permutation of the variables. Given a finite set Λ⊂Γ, we set H(Λ)=⊗i∈ΛH(i). An element ϕ∈H(Λ) is a complex function:
(5)xΛ*∈M*Λ⟼ϕ(xΛ*).
Here xΛ* is a collection {x*(j),j∈Λ} of finite point sets x*(j)⊂M associated with sites j∈Λ. Following [1], we call x*(j) a particle configuration at site j (which can be empty) and xΛ* a particle configuration in, or over, Λ. The space M*Λ of particle configurations in Λ can be represented as the Cartesian product (M*)×Λ where M* is the disjoint union ⋃k=0,1,…‍M(k) and M(k) is the collection of (unordered) k-point subsets of M. (One can consider M(k) as the factor of the “off-diagonal” set M≠k in the Cartesian power Mk under the equivalence relation induced by the permutation group of order k.) The norm and the scalar product in HΛ are given by
(6)∥ϕ∥=(∫M*Λ|ϕ(xΛ*)|2dxΛ*)1/2,〈ϕ1,ϕ2〉=∫M*Λϕ1(xΛ*)ϕ2(xΛ*)¯dxΛ*,
where measure dxΛ* is the product ×j∈Λdx*(j) and dx*(j) is the Poissonian sum measure on M*:
(7)dx*(j)=∑k=0,1,…1(♯x*(j)=k)1k!∏x∈x*(j)dv(x)e-v(M).
Here v(M) is the volume of torus M.As in [1], we assume that an action
(8)(g,x)∈G×M⟼gx∈M
is given, of a group G that is a Euclidean space or a torus of dimension d′≤d. The action is written as
(9)gx=x+θAmod1.
Here vector θ=(θ1,…,θd′) with components θl∈[0,1) and θA is the d-dimensional vector θ_=((θA)1,…,(θA)d) representing the element g, where A is a (d′×d) matrix of column rank d′ with rational entries. The action of G is lifted to unitary operators UΛ(g) in HΛ:
(10)UΛ(g)ϕ(xΛ*)=ϕ(g-1xΛ*),
where g-1xΛ*={g-1x*(j),j∈Λ} and g-1x*(j)={g-1x,x∈x*(j)}.The generally accepted view is that the Hubbard model is a highly oversimplified model for strongly interacting electrons in a solid. The Hubbard model is a kind of minimum model which takes into account quantum mechanical motion of electrons in a solid, and nonlinear repulsive interaction between electrons. There is little doubt that the model is too simple to describe actual solids faithfully [5]. In our context the Hubbard Hamiltonian HΛ of the system in Λ acts as follows:
(11)(HΛϕ)(xΛ*)=[∑x∈x*(j),x′∈x*(j′)V(x,x′)-12∑j∈Λ∑x∈x*(j)Δj(x)+∑j∈Λ‍∑x∈x*(j)U(1)(x)+12∑j∈Λ∑x,x′∈x*(j)1(x≠x′)U(2)(x,x′)+12∑j,j′∈Λ1(j≠j′)J(d(j,j′))×∑x∈x*(j),x′∈x*(j′)V(x,x′)]ϕ(xΛ*)+∑j,j′∈Λ‍λj,j′1(♯x*(j)≥1,♯x*(j′)<κ)×∑x∈x*(j)∫Mv(dy)×[ϕ(xΛ*(j,x)→(j′,y))-ϕ(xΛ*)].
Here Δj(x) means the Laplacian in variable x∈x*(j). Next, ♯x* stands for the cardinality of the particle configuration x* (i.e., ♯x*=k when x*∈M(k)), and the parameter κ is introduced in (17). (Symbol ♯ will be used for denoting the cardinality of a general (finite) set; for example, ♯Λ means the number of vertices in Λ.) Further, xΛ*(j,x)→(j′,y) denotes the particle configuration with the point x∈x*(j) removed and point y added to x*(j′).As in [1], we also consider a Hamiltonian HΛ∣x¯Γ¯∖Λ* in an external field generated by a configuration x¯Γ¯∖Λ*={x¯*(j′),j′∈Γ¯∖Λ}∈M*Γ¯∖Λ where Γ¯⊆Γ is a (finite or infinite) collection of vertices. More precisely, we only consider x¯Γ¯∖Λ* with ♯x¯*(j′)≤κ (see (17) below) and set
(12)(HΛ∣x¯Γ¯∖Λ*ϕ)(xΛ*)=[∑x∈x*(j),x′∈x*(j′)V(x,x′)-12∑j∈Λ∑x∈x*(j)Δj(x)+∑j∈Λ∑x∈x*(j)U(1)(x)+12∑j∈Λ∑x,x′∈x*(j)1(x≠x′)U(2)(x,x′)+12∑j,j′∈Λ‍1(j≠j′)J(d(j,j′))×∑x∈x*(j),x′∈x*(j′)V(x,x′)]ϕ(xΛ*)+∑j∈Λ,j′∈Γ¯∖Λ‍J(d(j,j′))×∑x∈x*(j),x-′∈x¯*(j′)V(x,x-′)ϕ(xΛ*)+∑j,j′∈Λ‍λj,j′1(♯x*(j)≥1,♯x*(j′)<κ)×∑x∈x*(j)∫Mv(dy)[ϕ(xΛ*(j,x)→(j′,y))-ϕ(xΛ*)].The novel elements in (11) and (12) compared with [1] are the presence of on-site potentials U(1) and U(2) and the summand involving transition rates λj,j′≥0 for jumps of a particle from site j to j′.We will suppose thatλj,j′ vanishes if the graph distance d(j,j′)>1. We will also assume uniform boundedness:
(13)sup[λj,j′(x,M),j,j′∈Γ,x∈M]<∞;
in view of (1) it implies that the total exit rate ∑j′:d(j,j′)=1λj,j′(x,M) from site j is uniformly bounded. These conditions are not sharp and can be liberalized.The model under consideration can be considered as a generalization of the Hubbard model [6] (in its bosonic version). Its mathematical justification includes the following. (a) An opportunity to introduce a Fock space formalism incorporates a number of new features. For instance, a fermonic version of the model (not considered here) emerges naturally when the bosonic Fock space H(i) is replaced by a fermonic one. Another opening provided by this model is a possibility to consider random potentials U(1), U(2) and V which would yield a sound generalization of the Mott-Anderson model. (b) Introducing jumps makes a step towards a treatment of a model of a quantum (Bose-) gas where particles “live” in a single Fock space. For example, a system of interacting quantum particles is originally confined to a “box” in a Euclidean space, with or without “internal” degrees of freedom. In the thermodynamical limit the box expands to the whole Euclidean space. In a two-dimensional model of a quantum gas one expects a phenomenon of invariance under space-translations; one hopes to be able to address this issue in future publications. (c) A model with jumps can be analysed by means of the theory of Markov processes which provides a developed methodology.Physically speaking, the model with jumps covers a situation where “light” quantum particles are subject to a “random” force and change their “location.” This class of models is interesting from the point of view of transport phenomena that they may display. (An analogy with the famous Anderson model, in its multiparticle version, inevitably comes to mind; cf., e.g., [7].) Methodologically, such systems occupy an “intermediate” place between models where quantum particles are “fixed” forever to their designated locations (as in [1]) and models where quantum particles move in the same space (a Bose-gas, considered in [8, 9]). In particular, this work provides a bridge between [1, 8, 9]; reading this paper ahead of [8, 9] might help an interested reader to get through [8, 9] at a much quicker pace.We would like to note an interesting problem of analysis of the small-mass limit (cf. [10]) from the point of Mermin-Wagner phenomena.


### 1.3. Assumptions on the Potentials

The between-sites potentialV is assumed to be of class C2. Consequently, V and its first and second derivatives satisfy uniform bounds. Namely, ∀x′,x′′∈M(14)-V(x′,x′′),|∇xV(x′,x′′)|,|∇x,x′V(x′,x′′)|≤V¯.
Here x and x′ run through the pairs of variables x, x′. A similar property is assumed for the on-site potential U(1) (here we need only a C1 smoothness):
(15)-U(1)(x),|∇xU(1)(x)|≤U¯(1),x∈M.
Note that for V and U(1) the bounds are imposed on their negative parts only.As toU(2), we suppose that (a)
(16)U(2)(x,x′)=+∞when|x-x′|≤ρ,
and (b) ∃ a C1-function (x,x′)↦U~(2)(x,x′)∈ℝ such that U(2)(x,x′)=U~(2)(x,x′) whenever ρ(x,x′)>ρ. Here ρ(x,x′) stands for the (flat) Riemannian distance between points x,x′∈M. As a result of (16), there exists a “hard core” of diameter ρ, and a given atom cannot “hold” more than
(17)κ=⌈v(M)v(B(ρ))⌉
particles where v(B(ρ)) is the volume of a d-dimensional ball of diameter ρ. We will also use the bound
(18)-U~(2)(x,x′),|∇xU~(2)(x,x′)|≤U¯(2),x,x′∈M.
Formally, (16) means that the operators in (11) and (12) are considered for functions ϕ(xΛ*) vanishing when in the particle configuration xΛ*={x*(j),j∈Λ}, the cardinality ♯x*(j)>κ for some j∈Λ.The functionJ:r∈(0,∞)↦J(r)≥0 is assumed monotonically nonincreasing with r and obeying the relation J¯(l)→0 as l→∞, where
(19)J¯(l)=sup[∑j′′∈Γ‍J(d(j′,j′′))1(d(j′,j′′)≥l):j′∈Γ]<∞.
Additionally, let J(r) be such that
(20)J*=sup[∑j′∈ΓJ(d(j,j′))d(j,j′)2:j∈Γ]<∞.
Next, we assume that the functions U(1), U(2), and V are g-invariant: ∀x,x′∈M and g∈G,
(21)U(1)(x)=U(1)(gx),U(2)(x,x′)=U(2)(gx,gx′),V(x,x′)=V(gx,gx′).
In the following we will need to bound the fugacity (or activity, cf. (25)) z in terms of the other parameters of the model
(22)zeΘ<1,whereΘ=κβ(U¯(1)+κU¯(2)+κJ¯(1)V¯).


### 1.4. The Gibbs State in a Finite Volume

Define the particle number operatorNΛ, with the action
(23)NΛϕ(xΛ*)=♯xΛ*ϕ(xΛ*),xΛ*∈M*Λ.
Here, for a given xΛ*={x*(j),j∈Λ}, ♯xΛ* stands for the total number of particles in configuration xΛ*:
(24)♯xΛ*=∑j∈Λ‍♯x*(j).
The standard canonical variable associated with NΛ is activity z∈(0,∞).The Hamiltonians (11) and (12) are self-adjoint (on the natural domains) in H(Λ). Moreover, they are positive definite and have a discrete spectrum, cf. [14]. Furthermore, ∀z,β>0, HΛ and HΛ∣x¯Γ¯∖Λ* give rise to positive-definite trace-class operators GΛ=Gz,β,Λ and GΛ∣x¯Γ¯∖Λ*=Gz,β,Λ∣x¯Γ¯∖Λ*:
(25)GΛ=zNΛexp[-βHΛ],GΛ∣x¯Γ¯∖Λ*=zNΛexp[-βHΛ∣x¯Γ¯∖Λ*].
We would like to stress that the full range of variables z,β>0 is allowed here because of the hard-core condition (16): it does not allow more than κ♯Λ particles in Λ where ♯Λ stands for the number of vertices in Λ. However, while passing to the thermodynamic limit, we will need to control z and β.Definition 1.
We will callGΛ and GΛ∣x¯Γ¯∖Λ* the Gibbs operators in volume Λ, for given values of z and β (and—in the case of GΛ∣x¯Γ¯∖Λ*—with the boundary condition x¯Γ¯∖Λ*).
The Gibbs operators in turn give rise to the Gibbs statesφΛ=φβ,z,Λ and φΛ∣x¯Γ¯∖Λ=φβ,z,Λ∣x¯Γ¯∖Λ, at temperature β-1 and activity z in volume Λ. These are linear positive normalized functionals on the C*-algebra BΛ of bounded operators in space HΛ:
(26)φΛ(A)=trHΛ(RΛA),φΛ∣x¯Γ¯∖Λ(A)=trHΛ(RΛ∣x¯Γ¯∖ΛA),A∈BΛ,
where
(27)RΛ=GΛΞ(Λ),withΞ(Λ)=Ξz,β(Λ)=trHΛGΛ,(28)RΛ∣x¯Γ¯∖Λ*=GΛ∣x¯Γ¯∖ΛΞ(Λ∣x¯Γ¯∖Λ*),withΞ(Λ∣x¯Γ¯∖Λ*)=Ξz,β(Λ∣x¯Γ¯∖Λ*)withΞ(Λ∣x¯Γ¯∖Λ*)=trHΛ(zNΛexp[-βHΛ∣x¯Γ¯∖Λ*]).The hard-core assumption (16) yields that the quantities Ξ(Λ) and Ξz,β(Λ∣x¯Γ¯∖Λ*) are finite; formally, these facts will be verified by virtue of the Feynman-Kac representation.Definition 2.
WheneverΛ0⊂Λ, the C*-algebra BΛ0 is identified with the C* subalgebra in BΛ formed by the operators of the form A0⊗IΛ∖Λ0. Consequently, the restriction φΛΛ0 of state φΛ to C*-algebra BΛ0 is given by
(29)φΛΛ0(A0)=trHΛ0(RΛΛ0A0),A0∈BΛ0,
where
(30)RΛΛ0=trHΛ∖Λ0RΛ.
Operators RΛΛ0 (we again call them RDMs) are positive definite and have trHΛ0RΛΛ0=1. They also satisfy the compatibility property: ∀Λ0⊂Λ1⊂Λ,
(31)RΛΛ0=trHΛ1∖Λ0RΛΛ1.
In a similar fashion one defines functionals φΛ∣x¯Γ¯∖Λ*Λ0 and operators RΛ∣x¯Γ¯∖Λ*Λ0, with the same properties.


### 1.5. Limiting Gibbs States

The concept of a limiting Gibbs state is related to notion of a quasilocalC*-algebra; see [14]. For the class of systems under consideration, the construction of the quasilocal C*-algebra BΓ is done along the same lines as in [1]: BΓ is the norm completion of the C* algebra (BΓ0)=limindn→∞BΛn. Any family of positive-definite operators RΛ0 in spaces HΛ0 of trace one, where Λ0 runs through finite subsets of Γ, with the compatibility property
(32)RΛ1=trHΛ0∖Λ1RΛ0,Λ1⊂Λ0,
determines a state of BΓ, see [12, 13].Finally, we introduce unitary operatorsUΛ0(g), g∈G, in HΛ0:
(33)UΛϕ(xΛ0*)=ϕ(g-1xΛ0*),
where
(34)g-1xΛ0*={g-1x*(j),j∈Λ0},g-1x*(j)={g-1x:x∈x*(j)}.Theorem 3.
Assuming the conditions listed above, for allz,β∈(0,+∞) satisfying (22) and a finite Λ0⊂Γ, operators RΛΛ0 form a compact sequence in the trace-norm topology in HΛ0 as Λ↗Γ. Furthermore, given any family of (finite or infinite) sets Γ¯=Γ¯(Λ)⊆Γ and configurations x¯Γ¯∖Λ*={x¯*(i),i∈Γ¯∖Λ} with ♯x¯*(i)<κ, operators RΛ∣x¯Γ¯∖ΛΛ0 also form a compact sequence in the trace-norm topology. Any limit point, RΛ0, for {RΛΛ0} or {RΛ∣x¯Γ¯∖Λ*Λ0} as Λ↗Γ, is a positive-definite operator in ℋ(Λ0) of trace one. Moreover, if Λ1⊂Λ0 and RΛ0 and RΛ1 are the limits for RΛΛ0 and RΛΛ1 or for RΛ∣x¯Γ¯∖Λ*Λ0 and RΛ∣x¯Γ¯∖Λ*Λ1 along the same subsequence Λs↗Γ, then the property (32) holds true.
Consequently, the Gibbs statesφΛ and φΛ∣x¯Γ¯∖Λ* form compact sequences as Λ↗Γ.Remark 4.
In fact, the assertion of Theorem3 holds without assuming the bidimensionality condition on graph (Γ,ℰ), only under an assumption that the degree of the vertices in Γ is uniformy bounded.Definition 5.
Any limit pointφ for states φΛ and φΛ∣x¯Γ¯∖Λ* is called a limiting Gibbs state (for given z,β∈(0,+∞)).Theorem 6.
Under the condition (22), any limiting point, RΛ0, for {RΛΛ0} or {RΛ∣x¯Γ¯∖Λ*Λ0}, as Λ↗Γ, is a positive-definite operator of trace one commuting with operators UΛ0(g): ∀g∈G,
(35)UΛ0(g)-1RΛ0UΛ0(g)=RΛ0.
Accordingly, any limiting Gibbs state φ of B determined by a family of limiting operators RΛ0 obeying (35) satisfies the corresponding invariance property: ∀ finite Λ0⊂Γ, any A∈BΛ0, and g∈G,
(36)φ(A)=φ(UΛ0(g)-1AUΛ0(g)).Remarks. (1) Condition (22) does not imply the uniqueness of an infinite-volume Gibbs state (i.e., absence of phase transitions).(2) Properties (35) and (36) are trivially fulfilled for the limiting points RΛ0 and φ of families {RΛΛ0} and {φΛ}. However, they require a proof for the limit points of the families {RΛ∣x¯Γ∖Λ*Λ0} and {φΛ∣x¯Γ∖Λ*}.The set of limiting Gibbs states (which is nonempty due to Theorem3) is denoted by G0. In Section 3 we describe a class G⊃G0 of states of C*-algebra B satisfying the FK-DLR equation, similar to that in [1].


## 1.1. Basic Facts on Bi-Dimensional Graphs

As in [1], we suppose that the graph (Γ,ℰ) has been given, with the set of vertices Γ and the set of edges ℰ. The graph has the property that whenever edge (j′,j′′)∈ℰ, the reversed edge (j′′,j′) belongs to ℰ as well. Furthermore, graph (Γ,ℰ) is without multiple edges and has a bounded degree; that is, the number of edges (j,j′) with a fixed initial or terminal vertex is uniformly bounded: (1)sup[max(♯{j′∈Γ:(j,j′)∈ℰ},supamx5555♯{j′∈Γ:(j′,j)∈ℰ}):j∈Γ]<∞.
The bi-dimensionality property is expressed in the bound
(2)0<sup[1n♯Σ(j,n):j∈Γ,n=1,2,…]<∞,
where Σ(j,n) stands for the set of vertices in Γ at the graph distance n from j∈Γ (a sphere of radius n around j):
(3)Σ(j,n)={j′∈Γ:d(j,j′)=n}.
(The graph distance d(j,j′)=dΓ,ℰ(j,j′) between j,j′∈Γ is determined as the minimal length of a path on (Γ,ℰ) joining j and j′.) This implies that for any o∈Γ the cardinality ♯Λ(o,n) of the ball
(4)Λ(o,n)={j′∈Γ:d(o,j′)≤n}
grows at most quadratically with n.A justification for putting a quantum system on a graph can be that graph-like structures become increasingly popular in rigorous Statistical Mechanics, for example, in quantum gravity. Namely, see [2–4]. On the other hand, a number of properties of Gibbs ensembles do not depend upon “regularity” of an underlying spatial geometry.


## 1.2. A Bosonic Model in the Fock Space

With each vertexi∈Γ we associate a copy of a compact manifold M which we take in this paper to be a unit d-dimensional torus ℝd/ℤd with a flat metric ρ and the volume v. We also associate with i∈Γ a bosonic Fock-Hilbert space H(i)≃H. Here H=⊕k=0,1,…ℋk where ℋk=L2sym(Mk) is the subspace in L2(Mk) formed by functions symmetric under a permutation of the variables. Given a finite set Λ⊂Γ, we set H(Λ)=⊗i∈ΛH(i). An element ϕ∈H(Λ) is a complex function:
(5)xΛ*∈M*Λ⟼ϕ(xΛ*).
Here xΛ* is a collection {x*(j),j∈Λ} of finite point sets x*(j)⊂M associated with sites j∈Λ. Following [1], we call x*(j) a particle configuration at site j (which can be empty) and xΛ* a particle configuration in, or over, Λ. The space M*Λ of particle configurations in Λ can be represented as the Cartesian product (M*)×Λ where M* is the disjoint union ⋃k=0,1,…‍M(k) and M(k) is the collection of (unordered) k-point subsets of M. (One can consider M(k) as the factor of the “off-diagonal” set M≠k in the Cartesian power Mk under the equivalence relation induced by the permutation group of order k.) The norm and the scalar product in HΛ are given by
(6)∥ϕ∥=(∫M*Λ|ϕ(xΛ*)|2dxΛ*)1/2,〈ϕ1,ϕ2〉=∫M*Λϕ1(xΛ*)ϕ2(xΛ*)¯dxΛ*,
where measure dxΛ* is the product ×j∈Λdx*(j) and dx*(j) is the Poissonian sum measure on M*:
(7)dx*(j)=∑k=0,1,…1(♯x*(j)=k)1k!∏x∈x*(j)dv(x)e-v(M).
Here v(M) is the volume of torus M.As in [1], we assume that an action
(8)(g,x)∈G×M⟼gx∈M
is given, of a group G that is a Euclidean space or a torus of dimension d′≤d. The action is written as
(9)gx=x+θAmod1.
Here vector θ=(θ1,…,θd′) with components θl∈[0,1) and θA is the d-dimensional vector θ_=((θA)1,…,(θA)d) representing the element g, where A is a (d′×d) matrix of column rank d′ with rational entries. The action of G is lifted to unitary operators UΛ(g) in HΛ:
(10)UΛ(g)ϕ(xΛ*)=ϕ(g-1xΛ*),
where g-1xΛ*={g-1x*(j),j∈Λ} and g-1x*(j)={g-1x,x∈x*(j)}.The generally accepted view is that the Hubbard model is a highly oversimplified model for strongly interacting electrons in a solid. The Hubbard model is a kind of minimum model which takes into account quantum mechanical motion of electrons in a solid, and nonlinear repulsive interaction between electrons. There is little doubt that the model is too simple to describe actual solids faithfully [5]. In our context the Hubbard Hamiltonian HΛ of the system in Λ acts as follows:
(11)(HΛϕ)(xΛ*)=[∑x∈x*(j),x′∈x*(j′)V(x,x′)-12∑j∈Λ∑x∈x*(j)Δj(x)+∑j∈Λ‍∑x∈x*(j)U(1)(x)+12∑j∈Λ∑x,x′∈x*(j)1(x≠x′)U(2)(x,x′)+12∑j,j′∈Λ1(j≠j′)J(d(j,j′))×∑x∈x*(j),x′∈x*(j′)V(x,x′)]ϕ(xΛ*)+∑j,j′∈Λ‍λj,j′1(♯x*(j)≥1,♯x*(j′)<κ)×∑x∈x*(j)∫Mv(dy)×[ϕ(xΛ*(j,x)→(j′,y))-ϕ(xΛ*)].
Here Δj(x) means the Laplacian in variable x∈x*(j). Next, ♯x* stands for the cardinality of the particle configuration x* (i.e., ♯x*=k when x*∈M(k)), and the parameter κ is introduced in (17). (Symbol ♯ will be used for denoting the cardinality of a general (finite) set; for example, ♯Λ means the number of vertices in Λ.) Further, xΛ*(j,x)→(j′,y) denotes the particle configuration with the point x∈x*(j) removed and point y added to x*(j′).As in [1], we also consider a Hamiltonian HΛ∣x¯Γ¯∖Λ* in an external field generated by a configuration x¯Γ¯∖Λ*={x¯*(j′),j′∈Γ¯∖Λ}∈M*Γ¯∖Λ where Γ¯⊆Γ is a (finite or infinite) collection of vertices. More precisely, we only consider x¯Γ¯∖Λ* with ♯x¯*(j′)≤κ (see (17) below) and set
(12)(HΛ∣x¯Γ¯∖Λ*ϕ)(xΛ*)=[∑x∈x*(j),x′∈x*(j′)V(x,x′)-12∑j∈Λ∑x∈x*(j)Δj(x)+∑j∈Λ∑x∈x*(j)U(1)(x)+12∑j∈Λ∑x,x′∈x*(j)1(x≠x′)U(2)(x,x′)+12∑j,j′∈Λ‍1(j≠j′)J(d(j,j′))×∑x∈x*(j),x′∈x*(j′)V(x,x′)]ϕ(xΛ*)+∑j∈Λ,j′∈Γ¯∖Λ‍J(d(j,j′))×∑x∈x*(j),x-′∈x¯*(j′)V(x,x-′)ϕ(xΛ*)+∑j,j′∈Λ‍λj,j′1(♯x*(j)≥1,♯x*(j′)<κ)×∑x∈x*(j)∫Mv(dy)[ϕ(xΛ*(j,x)→(j′,y))-ϕ(xΛ*)].The novel elements in (11) and (12) compared with [1] are the presence of on-site potentials U(1) and U(2) and the summand involving transition rates λj,j′≥0 for jumps of a particle from site j to j′.We will suppose thatλj,j′ vanishes if the graph distance d(j,j′)>1. We will also assume uniform boundedness:
(13)sup[λj,j′(x,M),j,j′∈Γ,x∈M]<∞;
in view of (1) it implies that the total exit rate ∑j′:d(j,j′)=1λj,j′(x,M) from site j is uniformly bounded. These conditions are not sharp and can be liberalized.The model under consideration can be considered as a generalization of the Hubbard model [6] (in its bosonic version). Its mathematical justification includes the following. (a) An opportunity to introduce a Fock space formalism incorporates a number of new features. For instance, a fermonic version of the model (not considered here) emerges naturally when the bosonic Fock space H(i) is replaced by a fermonic one. Another opening provided by this model is a possibility to consider random potentials U(1), U(2) and V which would yield a sound generalization of the Mott-Anderson model. (b) Introducing jumps makes a step towards a treatment of a model of a quantum (Bose-) gas where particles “live” in a single Fock space. For example, a system of interacting quantum particles is originally confined to a “box” in a Euclidean space, with or without “internal” degrees of freedom. In the thermodynamical limit the box expands to the whole Euclidean space. In a two-dimensional model of a quantum gas one expects a phenomenon of invariance under space-translations; one hopes to be able to address this issue in future publications. (c) A model with jumps can be analysed by means of the theory of Markov processes which provides a developed methodology.Physically speaking, the model with jumps covers a situation where “light” quantum particles are subject to a “random” force and change their “location.” This class of models is interesting from the point of view of transport phenomena that they may display. (An analogy with the famous Anderson model, in its multiparticle version, inevitably comes to mind; cf., e.g., [7].) Methodologically, such systems occupy an “intermediate” place between models where quantum particles are “fixed” forever to their designated locations (as in [1]) and models where quantum particles move in the same space (a Bose-gas, considered in [8, 9]). In particular, this work provides a bridge between [1, 8, 9]; reading this paper ahead of [8, 9] might help an interested reader to get through [8, 9] at a much quicker pace.We would like to note an interesting problem of analysis of the small-mass limit (cf. [10]) from the point of Mermin-Wagner phenomena.


## 1.3. Assumptions on the Potentials

The between-sites potentialV is assumed to be of class C2. Consequently, V and its first and second derivatives satisfy uniform bounds. Namely, ∀x′,x′′∈M(14)-V(x′,x′′),|∇xV(x′,x′′)|,|∇x,x′V(x′,x′′)|≤V¯.
Here x and x′ run through the pairs of variables x, x′. A similar property is assumed for the on-site potential U(1) (here we need only a C1 smoothness):
(15)-U(1)(x),|∇xU(1)(x)|≤U¯(1),x∈M.
Note that for V and U(1) the bounds are imposed on their negative parts only.As toU(2), we suppose that (a)
(16)U(2)(x,x′)=+∞when|x-x′|≤ρ,
and (b) ∃ a C1-function (x,x′)↦U~(2)(x,x′)∈ℝ such that U(2)(x,x′)=U~(2)(x,x′) whenever ρ(x,x′)>ρ. Here ρ(x,x′) stands for the (flat) Riemannian distance between points x,x′∈M. As a result of (16), there exists a “hard core” of diameter ρ, and a given atom cannot “hold” more than
(17)κ=⌈v(M)v(B(ρ))⌉
particles where v(B(ρ)) is the volume of a d-dimensional ball of diameter ρ. We will also use the bound
(18)-U~(2)(x,x′),|∇xU~(2)(x,x′)|≤U¯(2),x,x′∈M.
Formally, (16) means that the operators in (11) and (12) are considered for functions ϕ(xΛ*) vanishing when in the particle configuration xΛ*={x*(j),j∈Λ}, the cardinality ♯x*(j)>κ for some j∈Λ.The functionJ:r∈(0,∞)↦J(r)≥0 is assumed monotonically nonincreasing with r and obeying the relation J¯(l)→0 as l→∞, where
(19)J¯(l)=sup[∑j′′∈Γ‍J(d(j′,j′′))1(d(j′,j′′)≥l):j′∈Γ]<∞.
Additionally, let J(r) be such that
(20)J*=sup[∑j′∈ΓJ(d(j,j′))d(j,j′)2:j∈Γ]<∞.
Next, we assume that the functions U(1), U(2), and V are g-invariant: ∀x,x′∈M and g∈G,
(21)U(1)(x)=U(1)(gx),U(2)(x,x′)=U(2)(gx,gx′),V(x,x′)=V(gx,gx′).
In the following we will need to bound the fugacity (or activity, cf. (25)) z in terms of the other parameters of the model
(22)zeΘ<1,whereΘ=κβ(U¯(1)+κU¯(2)+κJ¯(1)V¯).


## 1.4. The Gibbs State in a Finite Volume

Define the particle number operatorNΛ, with the action
(23)NΛϕ(xΛ*)=♯xΛ*ϕ(xΛ*),xΛ*∈M*Λ.
Here, for a given xΛ*={x*(j),j∈Λ}, ♯xΛ* stands for the total number of particles in configuration xΛ*:
(24)♯xΛ*=∑j∈Λ‍♯x*(j).
The standard canonical variable associated with NΛ is activity z∈(0,∞).The Hamiltonians (11) and (12) are self-adjoint (on the natural domains) in H(Λ). Moreover, they are positive definite and have a discrete spectrum, cf. [14]. Furthermore, ∀z,β>0, HΛ and HΛ∣x¯Γ¯∖Λ* give rise to positive-definite trace-class operators GΛ=Gz,β,Λ and GΛ∣x¯Γ¯∖Λ*=Gz,β,Λ∣x¯Γ¯∖Λ*:
(25)GΛ=zNΛexp[-βHΛ],GΛ∣x¯Γ¯∖Λ*=zNΛexp[-βHΛ∣x¯Γ¯∖Λ*].
We would like to stress that the full range of variables z,β>0 is allowed here because of the hard-core condition (16): it does not allow more than κ♯Λ particles in Λ where ♯Λ stands for the number of vertices in Λ. However, while passing to the thermodynamic limit, we will need to control z and β.Definition 1.
We will callGΛ and GΛ∣x¯Γ¯∖Λ* the Gibbs operators in volume Λ, for given values of z and β (and—in the case of GΛ∣x¯Γ¯∖Λ*—with the boundary condition x¯Γ¯∖Λ*).
The Gibbs operators in turn give rise to the Gibbs statesφΛ=φβ,z,Λ and φΛ∣x¯Γ¯∖Λ=φβ,z,Λ∣x¯Γ¯∖Λ, at temperature β-1 and activity z in volume Λ. These are linear positive normalized functionals on the C*-algebra BΛ of bounded operators in space HΛ:
(26)φΛ(A)=trHΛ(RΛA),φΛ∣x¯Γ¯∖Λ(A)=trHΛ(RΛ∣x¯Γ¯∖ΛA),A∈BΛ,
where
(27)RΛ=GΛΞ(Λ),withΞ(Λ)=Ξz,β(Λ)=trHΛGΛ,(28)RΛ∣x¯Γ¯∖Λ*=GΛ∣x¯Γ¯∖ΛΞ(Λ∣x¯Γ¯∖Λ*),withΞ(Λ∣x¯Γ¯∖Λ*)=Ξz,β(Λ∣x¯Γ¯∖Λ*)withΞ(Λ∣x¯Γ¯∖Λ*)=trHΛ(zNΛexp[-βHΛ∣x¯Γ¯∖Λ*]).The hard-core assumption (16) yields that the quantities Ξ(Λ) and Ξz,β(Λ∣x¯Γ¯∖Λ*) are finite; formally, these facts will be verified by virtue of the Feynman-Kac representation.Definition 2.
WheneverΛ0⊂Λ, the C*-algebra BΛ0 is identified with the C* subalgebra in BΛ formed by the operators of the form A0⊗IΛ∖Λ0. Consequently, the restriction φΛΛ0 of state φΛ to C*-algebra BΛ0 is given by
(29)φΛΛ0(A0)=trHΛ0(RΛΛ0A0),A0∈BΛ0,
where
(30)RΛΛ0=trHΛ∖Λ0RΛ.
Operators RΛΛ0 (we again call them RDMs) are positive definite and have trHΛ0RΛΛ0=1. They also satisfy the compatibility property: ∀Λ0⊂Λ1⊂Λ,
(31)RΛΛ0=trHΛ1∖Λ0RΛΛ1.
In a similar fashion one defines functionals φΛ∣x¯Γ¯∖Λ*Λ0 and operators RΛ∣x¯Γ¯∖Λ*Λ0, with the same properties.


## 1.5. Limiting Gibbs States

The concept of a limiting Gibbs state is related to notion of a quasilocalC*-algebra; see [14]. For the class of systems under consideration, the construction of the quasilocal C*-algebra BΓ is done along the same lines as in [1]: BΓ is the norm completion of the C* algebra (BΓ0)=limindn→∞BΛn. Any family of positive-definite operators RΛ0 in spaces HΛ0 of trace one, where Λ0 runs through finite subsets of Γ, with the compatibility property
(32)RΛ1=trHΛ0∖Λ1RΛ0,Λ1⊂Λ0,
determines a state of BΓ, see [12, 13].Finally, we introduce unitary operatorsUΛ0(g), g∈G, in HΛ0:
(33)UΛϕ(xΛ0*)=ϕ(g-1xΛ0*),
where
(34)g-1xΛ0*={g-1x*(j),j∈Λ0},g-1x*(j)={g-1x:x∈x*(j)}.Theorem 3.
Assuming the conditions listed above, for allz,β∈(0,+∞) satisfying (22) and a finite Λ0⊂Γ, operators RΛΛ0 form a compact sequence in the trace-norm topology in HΛ0 as Λ↗Γ. Furthermore, given any family of (finite or infinite) sets Γ¯=Γ¯(Λ)⊆Γ and configurations x¯Γ¯∖Λ*={x¯*(i),i∈Γ¯∖Λ} with ♯x¯*(i)<κ, operators RΛ∣x¯Γ¯∖ΛΛ0 also form a compact sequence in the trace-norm topology. Any limit point, RΛ0, for {RΛΛ0} or {RΛ∣x¯Γ¯∖Λ*Λ0} as Λ↗Γ, is a positive-definite operator in ℋ(Λ0) of trace one. Moreover, if Λ1⊂Λ0 and RΛ0 and RΛ1 are the limits for RΛΛ0 and RΛΛ1 or for RΛ∣x¯Γ¯∖Λ*Λ0 and RΛ∣x¯Γ¯∖Λ*Λ1 along the same subsequence Λs↗Γ, then the property (32) holds true.
Consequently, the Gibbs statesφΛ and φΛ∣x¯Γ¯∖Λ* form compact sequences as Λ↗Γ.Remark 4.
In fact, the assertion of Theorem3 holds without assuming the bidimensionality condition on graph (Γ,ℰ), only under an assumption that the degree of the vertices in Γ is uniformy bounded.Definition 5.
Any limit pointφ for states φΛ and φΛ∣x¯Γ¯∖Λ* is called a limiting Gibbs state (for given z,β∈(0,+∞)).Theorem 6.
Under the condition (22), any limiting point, RΛ0, for {RΛΛ0} or {RΛ∣x¯Γ¯∖Λ*Λ0}, as Λ↗Γ, is a positive-definite operator of trace one commuting with operators UΛ0(g): ∀g∈G,
(35)UΛ0(g)-1RΛ0UΛ0(g)=RΛ0.
Accordingly, any limiting Gibbs state φ of B determined by a family of limiting operators RΛ0 obeying (35) satisfies the corresponding invariance property: ∀ finite Λ0⊂Γ, any A∈BΛ0, and g∈G,
(36)φ(A)=φ(UΛ0(g)-1AUΛ0(g)).Remarks. (1) Condition (22) does not imply the uniqueness of an infinite-volume Gibbs state (i.e., absence of phase transitions).(2) Properties (35) and (36) are trivially fulfilled for the limiting points RΛ0 and φ of families {RΛΛ0} and {φΛ}. However, they require a proof for the limit points of the families {RΛ∣x¯Γ∖Λ*Λ0} and {φΛ∣x¯Γ∖Λ*}.The set of limiting Gibbs states (which is nonempty due to Theorem3) is denoted by G0. In Section 3 we describe a class G⊃G0 of states of C*-algebra B satisfying the FK-DLR equation, similar to that in [1].


## 2. Feynman-Kac Representations for the RDM Kernels in a Finite Volume

### 2.1.  The Representation for the Kernels of the Gibbs Operators

A starting point for the forthcoming analysis is the Feynman-Kac (FK) representation for the kernelsKΛ(xΛ*,yΛ*)=Kβ,z,Λ(xΛ*,yΛ*) and FΛ(xΛ*,yΛ*)=Fβ,z,Λ(xΛ*,yΛ*) of operators GΛ and RΛ.Definition 7.
Given(x,i),(y,j)∈M×Γ, W¯(x,i),(y,j)β denotes the space of path, or trajectories, ω¯=ω¯(x,i),(y,j) in M×Γ, of time-length β, with the end-points (x,i) and (y,j). Formally, ω¯∈W¯(x,i),(y,j)β is defined as follows:
(37)ω¯:τ∈[0,β]⟼ω¯(τ)=(u(ω¯,τ),l(ω¯,τ))∈M×Γ,ω¯isca´dla´g;ω¯(0)=(x,i),ω¯(β-)=(y,j),ω¯hasfinitelymanyjumpson[0,β];ifajumpoccursattimeτ,thend[l(ω-,τ-),l(ω-,τ)]=1.
The notation ω¯(τ) and its alternative, (u(ω¯,τ),l(ω¯,τ)), for the position and the index of trajectory ω¯ at time τ will be employed as equal in rights. We use the term the temporal section (or simply the section) of path ω¯ at time τ.Definition 8.
LetxΛ*={x*(i),i∈Λ}∈M*Λ, and yΛ*={y*(j),j∈Λ}∈M*Λ be particle configurations over Λ, with ♯xΛ*=♯yΛ*. A matching (or pairing) γ between xΛ* and yΛ* is defined as a collection of pairs [(x,i),(y,j)]γ, with i,j∈Λ, x∈x*(i), and y∈y*(j), with the properties that (i) ∀i∈Λ and x∈x*(i): there exist unique j∈Λ and y∈y*(j) such that (x,i) and (y,j) form a pair, and (ii) ∀j∈Λ and y∈y*(j): there exist unique i∈Λ and x∈x*(i) such that (x,i) and (y,j) form a pair. (Owing to the condition ♯xΛ*=♯yΛ*, these properties are equivalent.) It is convenient to write [(x,i),(y,j)]γ=[(x,i),γ(x,i)].
Next, consider the Cartesian product(38)W¯xΛ*,yΛ*,γβ=×i∈Λ×x∈x*(i)W¯(x,i),γ(x,i)β,
and the disjoint union
(39)W¯xΛ*,yΛ*β=⋃‍γW¯xΛ*,yΛ*,γβ.
Accordingly, an element ω¯Λ∈W¯xΛ*,yΛ*,γβ in (38) represents a collection of paths ω¯x,i, x∈x*(i), i∈Λ, of time-length β, starting at (x,i) and ending up at γ(x,i). We say that ω¯Λ is a path configuration in (or over) Λ.The presence of matchings in the above construction is a feature of the bosonic nature of the systems under consideration.We will work with standard sigma algebras (generated by cylinder sets) inW¯(x,i),(y,j)β, W¯xΛ*,yΛ*,γβ, and W¯xΛ*,yΛ*β.Definition 9.
In what follows,ξ(τ), τ≥0, stands for the Markov process on M×Γ, with cádlág trajectories, determined by the generator ℒ acting on a function (x,i)∈M×Λ↦ψ(x,i) by
(40)ℒψ(x,i)=-12Δψ(x,i)ℒψ(x,i)=+∑j:d(i,j)=1λi,j∫Mv(dy)[ψ(y,j)-ψ(x,i)].
In the probabilistic literature, such processes are referred to as Lévy processes; see, for example, [14].Pictorially, a trajectory of processξ moves along M according to the Brownian motion with the generator -Δ/2 and changes the index i∈Γ from time to time in accordance with jumps occurring in a Poisson process of rate ∑j:d(i,j)=1λi.j. In other words, while following a Brownian motion rule on M, having index i∈Γ and being at point x∈M, the moving particle experiences an urge to jump from i to a neighboring vertex j and to a point y at rate λi,jv(dy). After a jump, the particle continues the Brownian motion on M from y and keeps its new index j until the next jump, and so on.For a given pairs(x,i),(y,j)∈M×Γ, we denote by ℙ(x,i),(y,j)β the nonnormalised measure on W¯(x,i),(y,j)β induced by ξ. That is, under measure ℙ(x,i),(y,j)β the trajectory at time τ=0 starts from the point x and has the initial index i while at time τ=β it is at the point y and has the index j. The value p^(x,i),(y,j)=ℙ(x,i),(y,j)β(W¯(x,i),(y,j)β) is given by
(41)p^(x,i),(y,j)=1(i=j)pMβ(x,y)exp[-β∑j~:d(i,j~)=1λi,j~]+∑k≥1∑l0=i,l1,…,lk,lk+1=j∏0≤s≤k1(d(ls,ls+1)=1)×λls,ls+1∫0βdτsexp[-(τs+1-τs)∑j~:d(ls,j~)=1λls,j~]×1(0=τ0<τ1<⋯<τk<τk+1=β),
where pMβ(x,y) denotes the transition probability density for the Brownian motion to pass from x to y on M in time β:
(42)pMβ(x,y)=1(2πβ)d/2×∑n_=(n1,…,nd)∈ℤdexp(-|x-y+n_|22β).
In view of (13), the quantity p^(x,i),(y,j) and its derivatives are uniformly bounded:
(43)p^(x,i),(y,j),|∇xp^(x,i),(y,j)|,|∇yp^(x,i),(y,j)|≤p^M,p^(x,i),(y,j),|∇xp^(x,i),(y,j)|1,x,y∈M,i,j∈Γ,
where p^M=p^M(β)∈(0,+∞) is a constant.We suggest a term “non-normalised Brownian bridge with jumps” for the measure but expect that a better term will be proposed in future.Definition 10.
Suppose thatxΛ*={x*(i),i∈Λ}∈M*Λ and yΛ*={y*(j),j∈Λ}∈M*Λ are particle configurations over Λ, with ♯xΛ*=♯yΛ*. Let γ be a pairing between xΛ* and yΛ*. Then ℙxΛ*,yΛ*,γ* denotes the product measure on W¯xΛ*,yΛ*,γβ:
(44)ℙ¯xΛ*,yΛ*,γβ=×i∈Λ×x∈x*(i)ℙ¯(x,i),γ(x,i)β.
Furthermore, ℙ¯xΛ*,yΛ*β stands for the sum measure on W¯xΛ*,yΛ*β:
(45)ℙ¯xΛ*,yΛ*β=∑γ‍ℙ¯xΛ*,yΛ*,γβ.According to Definition10, under the measure ℙ¯xΛ*,yΛ*,γβ, the trajectories ω¯x,i∈W¯(x,i),γ(x,i)β constituting ω¯Λ are independent components. (Here the term independence is used in the measure-theoretical sense.)As in [1], we will work with functionals on W¯xΛ*,yΛ*,γβ representing integrals along trajectories. The first such functional, hΛ(ω¯Λ), is given by
(46)hΛ(ω¯Λ)=∑i∈Λ∑x∈x*(i)hx,i(ω¯x,i)hΛ(ω¯Λ)=+12∑(i,i′)∈Λ×Λ∑x∈x*(i),x′∈x*(i′)h(x,i),(x′,i′)hΛ(ω¯Λ)=×(ω¯x,i,ω¯x′,i′).
Here, introducing the notation ux,i(τ)=u(ω¯x,i,τ) and ux′,i′(τ)=u(ω¯x′,i′,τ) for the positions in M of paths ω¯x,i∈W¯(x,i),γ(x,i)β and ω¯x′,i′∈W¯(x′,i′),γ(x′,i′)β at time τ, we define
(47)hx,i(ω¯x,i)=∫0βdτU(1)(ui,x(τ)).
Next, with lx,i(τ) and lx′,i′(τ) standing for the indices of ω¯x,i and ω¯x′,i′ at time τ,
(48)h(x,i),(x′,i′)(ω¯x,i,ω¯x′,i′)=∫0βdτ[∑j′∈ΓU(2)(ui,x(τ),ui′,x′(τ))×1(lx,i(τ)=j′=lx′,i′(τ))+12∑(j′,j′′)∈Γ×ΓJ(d(j′,j′′))×V(ui,x(τ),ui′,x′(τ))×1(lx,i(τ)=j′≠j′′=lx′,i′(τ))∑j′∈ΓU(2)(ui,x(τ),ux′,i′(τ))].Next, consider the functionalhΛ(ω¯Λ∣x¯Γ¯∖Λ*): for x¯Γ¯∖Λ*={x¯*(j),j∈Γ¯∖Λ}. As before, we assume that ♯x¯*(j)≤κ. Define
(49)hΛ(ω¯Λ∣x¯Γ¯∖Λ*)=hΛ(ω¯Λ)+hΛ(ω¯Λ∣∣x¯Γ¯∖Λ*).
Here hΛ(ω¯Λ) is as in (46) and
(50)hΛ(ω¯Λ∣∣x¯Γ¯∖Λ*)=∑(i,i′)∈Λ×(Γ¯∖Λ)∑x∈x*(i),x-′∈x¯*(i′)h(x,i),(x-′,i′)×(ω¯x,i,(x-′,i′)),
where, in turn,
(51)h(x,i),(x-′,i′)(ω¯x,i,(x-′,i′))=∫0βdτ[U(2)(ui,x(τ),x-′)1(lx,i(τ)=i′)+∑j∈Γ:j≠i′J(d(j,i′))×V(ui,x(τ),x-′)1(lx,i(τ)=j)].
The functionals hΛ(ω¯Λ) and hΛ(ω¯Λ∣∣x¯Γ¯∖Λ*) are interpreted as energies of path configurations. Compare (2.1.4) and (2.3.8) in [1].Finally, we introduce the indicator functionalαΛ(ω¯Λ):
(52)αΛ(ω¯Λ)={1,ifindexlx,i(τ)∈Λ,∀τ∈[0,β],i∈Λ,x∈x*(i),0,otherwise.
It can be derived from known results [11, 15–17] (for a direct argument, see [18]) that the following assertion holds true.Lemma 11.
For allz,β>0 and a finite Λ, the Gibbs operators GΛ and GΛ∣x¯Γ¯∖Λ* act as integral operators in ℋ(Λ):
(53)(GΛϕ)(xΛ*)=∫M*Λ∏j∈Λ∏y∈y*(j)v(dy)KΛ(xΛ*,yΛ*)ϕ(yΛ*),(GΛ∣x¯Γ¯∖Λ*ϕ)(xΛ*)=∫M*Λ∏j∈Λ∏y∈y*(j)v(dy)KΛ(xΛ*,yΛ*∣x¯Γ¯∖Λ*)ϕ(yΛ*).
Moreover, the integral kernels KΛ(xΛ*,yΛ*) and KΛ(xΛ*,yΛ*∣x¯Γ¯∖Λ*) vanish if ♯xΛ*≠♯yΛ*. On the other hand, when ♯xΛ*=♯yΛ*, the kernels KΛ(xΛ*,yΛ*) and KΛ(xΛ*,yΛ*∣x¯Γ¯∖Λ*) admit the following representations:
(54)KΛ(xΛ*,yΛ*)=z♯xΛ*∫W¯xΛ*,yΛ**ℙ¯xΛ*,yΛ*β(dω¯Λ)αΛ(ω¯Λ)×exp[-hΛ(ω¯Λ)],(55)KΛ(xΛ*,yΛ*∣x¯Γ¯∖Λ*)=z♯xΛ*∫W¯xΛ*,yΛ**ℙ¯xΛ*,yΛ*β(dω¯Λ)αΛ(ω¯Λ)×exp[-hΛ(ω¯Λ∣x¯Γ¯∖Λ*)].
The ingredients of these representations are determined in (46)–(51).Remark 12.
As before, we stress that, owing to (16) and (17), a nonzero contribution to the integral in the RHS of (54) can only come from a path configuration ω¯Λ={ω¯x,i} such that ∀τ∈[0,β] and ∀j∈Γ, the number of paths ω¯x,i with index lx,i(τ)=j is less than or equal to κ. Likewise, the integral in the RHS of (55) receives a non-zero contribution only from configurations ω¯Λ={ω¯x,i*} such that, ∀ site j∈Γ, the number of paths ω¯x,i with index lx,i(τ)=j plus the cardinality ♯x¯*(j) does not exceed κ.


### 2.2. The Representation for the Partition Function

The FK representations of the partition functionsΞ(Λ)=Ξβ,z(Λ) in (27) and Ξ(Λ∣x¯Γ¯∖Λ*) in (1.4.6) reflect a specific character of the traces trGΛ and trGΛ∣x¯Γ¯∖Λ* in ℋ(Λ). The source of a complication here is the jump terms in the Hamiltonians HΛ and HΛ∣x¯Γ¯∖Λ* in (11) and (12), respectively. In particular, we will have to pass from trajectories of fixed time-length β to loops of a variable time length. To this end, a given matching γ is decomposed into a product of cycles, and the trajectories associated with a given cycle are merged into closed paths (loops) of a time-length multiple of β. (A similar construction has been performed in [18].)To simplify the notation, we omit, wherever possible, the indexβ.Definition 13.
For given(x,i),(y,j)∈M×Γ, the symbol W¯(x,i),(y,j)* denotes the disjoint union:
(56)W¯(x,i),(y,j)*=⋃k=0,1,…W¯(x,i),(y,j)kβ.
In other words, W¯(x,i),(y,j)* is the space of paths Ω¯*=Ω¯(x,i),(y,j)* in M×Γ, of a variable time-length kβ, where k=k(Ω¯*) takes values 1,2,… and called the length multiplicity, with the end-points (x,i) and (y,j). The formal definition follows the same line as in (37), and we again use the notation Ω¯*(τ) and the notation (u(Ω¯*,τ),l(Ω¯*,τ)) for the pair of the position and the index of path Ω¯* at time τ. Next, we call the particle configuration {Ω¯*(τ+βm),0≤m<k(Ω¯*)} the temporal section (or simply the section) of Ω¯* at time τ∈[0,β]. We also call Ω¯(x,i),(y,j)*∈W¯(x,i),(y,j)* a path (from (x,i) to (y,j)).
A particular role will be played by closed paths (loops), with coinciding endpoints (where(x,i)=(y,j)). Accordingly, we denote by Wx,i* the set W¯(x,i),(x,i)*. An element of Wx,i* is denoted by Ωx,i* or, in short, by Ω* and called a loop at vertex i. (The upper index * indicates that the length multiplicity is unrestricted.) The length multiplicity of a loop Ωx,i*∈Wx,i* is denoted by k(Ωx,i*) or kx,i. It is instructive to note that, as topological object, a given loop Ω* admits a multiple choice of the initial pair (x,i): it can be represented by any pair (u(Ω*,τ),l(Ω*,τ)) at a time τ=lβ where l=1,…,k(Ω*). As above, we use the term the temporal section at time τ∈[0,β] for the particle configuration {Ωx,i*(τ+βm),0≤m<kx,i} and employ the alternative notation (u(τ+βm;Ω*),l(τ+βm;Ω*)) addressing the position and the index of Ω* at time τ+βm∈[0,βk(Ω*)].Definition 14.
SupposexΛ*={x*(i),i∈Λ}∈M*Λ and yΛ*={y*(j),j∈Λ}∈M*Λ are particle configurations over Λ, with ♯xΛ*=♯yΛ*. Let γ be a matching between xΛ* and yΛ*. We consider the Cartesian product:
(57)W¯xΛ*,yΛ*,γ*=×i∈Λ×x∈x*(i)W¯(x,i),γ(x,i)*,
and the disjoint union:
(58)W¯xΛ*,yΛ**=⋃γW¯xΛ*,yΛ*,γ*.
Accordingly, an element Ω-Λ*∈W¯xΛ*,yΛ*,γ* in (58) represents a collection of paths Ω-x,i*, x∈x*(i), i∈Λ, of time-length kβ, starting at (x,i) and ending up at (y,j)=γ(x,i). We say that Ω-Λ*∈W¯xΛ*,yΛ** is a path configuration in (or over) Λ.
Again, loops play a special role and deserve a particular notation. Namely,WxΛ** denotes the Cartesian product:
(59)WxΛ**=×i∈Λ×x∈x*(i)Wx,i*,
and WΛ* stands for the disjoint union (or equivalently, the Cartesian power):
(60)WΛ*=⋃xΛ*∈M*ΛWxΛ**=×i∈ΛW{i}*,whereW{i}*=⋃x*∈M*(×x∈x*Wx,i*).
Denote byΩ*={Ω*(i),i∈Λ}∈WΛ* a collection of loop configurations at vertices i∈Λ starting and ending up at particle configurations x*(i)∈M* (note that some of the Ω*(i)’s may be empty). The temporal section (or, in short, the section), Ω*(τ), of Ω* at time τ is defined as the particle configuration formed by the points Ωx,i*(τ+βm) where i∈Λ, x∈x*(i), and 0≤m<kx,i.As before, consider the standard sigma algebras of subsets in the spacesW¯(x,i),(y,j)*, Wx,i, W¯xΛ*,yΛ*,γ*, W¯xΛ*,yΛ**, WxΛ**, and WΛ* introduced in Definitions 13 and 14. In particular, the sigma algebra of subsets in WΛ* will be denoted by WΛ; we comment on some of its specific properties in Section 3.1. (An infinite-volume version WΓ* of WΛ* is treated in Section 3.2 and after.)Definition 15.
Given points(x,i),(y,j)∈M×Γ, we denote by ℙ¯(x,i),(y,j)* the sum measure on W¯(x,i),(y,j)*:
(61)ℙ¯(x,i),(y,j)*=∑k=0,1,…ℙ¯(x,i),(y,j)kβ.
Further,ℙx,i* denotes the similar measure on Wx,i*:
(62)ℙx,i*=∑k=0,1,…ℙx,ikβ.Definition 16.
LetxΛ*={x*(i),i∈Λ}∈M*Λ and yΛ*={y*(j),j∈Λ}∈M*Λ be particle configurations over Λ, with ♯xΛ*=♯yΛ*. Let γ be a matching between xΛ* and yΛ*, and we define the product measure ℙ¯xΛ*,yΛ*,γ*:
(63)ℙ¯xΛ*,yΛ*,γ*=×i∈Λ×x∈x*(i)ℙ¯(x,i),γ(x,i)*
and the sum measure
(64)ℙ¯xΛ*,yΛ**=∑γℙ¯xΛ*,yΛ*,γ*.
Next, symbolℙxΛ** stands for the product measure on WxΛ**:
(65)ℙxΛ**=×i∈Λ×x∈x*(i)ℙx,i*.
Finally,dΩΛ* yields the measure on WΛ*:
(66)dΩΛ*=dxΛ*×ℙxΛ**(dΩΛ*).
Here, for xΛ*={x*(i),i∈Λ}, we set: dxΛ*=∏i∈Λ∏x∈x*(i)v(dx). For sites i with x*(i)=∅, the corresponding factors are trivial measures sitting on the empty configurations.We again need to introduce energy-type functionals represented by integrals along loops. More precisely, we define the functionalshΛ(ΩΛ*) and hΛ(ΩΛ*∣x¯Γ¯∖Λ*) which are modifications of the above functionals hΛ(ω¯Λ) and hΛ(ω¯Λ∣x¯Γ¯∖Λ*); confer (46) and (50). Say, for a loop configuration ΩΛ*={Ωx,i*} over Λ with an initial and final particle configuration xΛ*={x*(i),i∈Λ},
(67)hΛ(ΩΛ*)=∑i∈Λ∑x∈x*(i)hx,i(Ωx,i*)+12∑(i,i′)∈Λ×Λ∑x∈x*(i),x′∈x*(i′)×1((x,i)≠(x′,i′))h(x,i),(x′,i′)(Ωx,i*,Ωx′,i′*).
To determine the functionals hx,i(Ωx,i*) and h(x,i),(x′,i′)(Ωx,i*,Ωx′,i′*), we set, for given m=0,1,…,kx,i-1 and m′=0,1,…,kx′,i′-1:
(68)ui,x(τ+βm)=u(τ+βm;Ωx,i*),li,x(τ+βm)=l(τ+βm;Ωx,i*),ui′,x′(τ+βm′)=u(τ+βm′;Ωx′,i′*),li′,x′(τ+βm′)=l(τ+βm′;Ωx′,i′*).
A (slightly) shortened notation li,x(τ+βm) is used for the index lx,i(τ+βm;Ωx,i*) and ui,x(τ+βm) for the position u(τ+βm;Ωx,i*) for Ωx,i*(τ)∈M×Γ, of the section Ωx,i*(τ) of the loop Ωx,i* at time τ, and similarly with li′,x′(τ+βm′) and ui′,x′(τ+βm′). (Note that the pairs (x,i) and (x′,i′) may coincide.) Then
(69)h(x,i)(Ωx,i*)=∫0βdτ[∑0≤m<kx,iU(1)(ux,i(τ+βm))+∑0≤m<m′<kx,i∑j∈Γ1(lx,i(τ+βm)=j=lx,i(τ+βm′))×U(2)(ux,i(τ+βm),ux,i(τ+βm′))∑0≤m<kx,iU(1)(ux,i(τ+βm))],h(x,i),(x′,i′)(Ωx,i*,Ωx′,i′*)=∑0≤m<kx,i∑0≤m′<kx′,i′∫0βdτ×[∑j∈ΓU(2)(ux,i(τ+βm),ux′,i′(τ+βm′))×1(lx,i(τ+βm)=j=lx′,i′(τ+βm′))+∑(j,j′)∈Γ×ΓJ(d(j,j′))V×(ui,x(τ+βm),ui′,x′(τ+βm′))×1(lx,i(τ+βm)=j≠j′=lx′,i′(τ+βm′))∑j∈ΓU(2)(ux,i(τ+βm),ux′,i′(τ+βm′))].Next, the functionalB(ΩΛ*) takes into account the bosonic character of the model:
(70)B(ΩΛ*)=∏i∈Λ∏x∈x*(i)zkx,ikx,i.
The factor kx,i-1 in (70) reflects the fact that the starting point of a loop Ωx,i* may be selected among points u(βm,Ωx,i*) arbitrarily.Next, we define the functionalhΛ(ΩΛ*∣x¯Γ¯∖Λ*): for x¯Γ¯∖Λ*={x¯*(j),j∈Γ¯∖Λ}, again assuming that ♯x¯*(j)≤κ. Set
(71)hΛ(ΩΛ*∣x¯Γ¯∖Λ*)=hΛ(ΩΛ*)+hΛ(ΩΛ*∣∣x¯Γ¯∖Λ*).
Here hΛ(ΩΛ*) is as in (67) and
(72)hΛ(ΩΛ*∣∣x¯Γ¯∖Λ*)=∑(i,i′)∈Λ×(Γ¯∖Λ)∑x∈x*(i),x-′∈x¯*(i′)h(x,i),(x-′,i′)(Ωx,i*,(x-′,i′)),
where, in turn,
(73)h(x,i),(x-′,i′)(Ωx,i*,(x-′,i′))=∑0≤m<kx,i‍∫0βdτ×[U(2)(ux,i(τ+βm),x′)×1(lx,i(τ+βm)=i′)+∑j∈ΓJ(d(j,i′))V(ui,x(τ+βm),x′)×1(lx,i(τ+βm)=j)U(2)(ux,i(τ+βm),x′)].
As before, the functionals hΛ(ΩΛ*) and hΛ(ΩΛ*∣∣x¯Γ¯∖Λ*) have a natural interpretation as energies of loop configurations.Finally, as before, the functionalαΛ(ΩΛ*) is the indicator that the collection of loops ΩΛ*={Ωx,i*} does not quit Λ:
(74)αΛ(ΩΛ*)={1,ifΩx,i*(τ)∈M×Λ,∀i∈Λ,x∈x*(i),0≤τ≤βkx,i,0,otherwise.Like above, we invoke known results [11, 15–17] to establish the following statement (again a direct argument can be found in [18]).Lemma 17.
For all finiteΛ⊂Γ and z,β>0 satisfying (22), the partition functions Ξ(Λ) in (27) and Ξ(Λ∣x¯Γ¯∖Λ*) in (27) and (28) admit the representations as converging integrals:
(75)Ξ(Λ)=∫WΛ*dΩΛ*B(ΩΛ*)αΛ(ΩΛ*)exp[-hΛ(ΩΛ*)],(76)Ξ(Λ∣x¯Γ¯∖Λ*)=∫WΛ*dΩΛ*B(ΩΛ*)αΛ(ΩΛ*)exp[-hΛ(ΩΛ*∣∣x¯Γ¯∖Λ*)]
with the ingredients introduced in (60)–(74).Again, we emphasis that the non-zero contribution to the integral in (75) can only come from loop configurations ΩΛ*={Ωx,i*,x∈x*(i),i∈Λ} such that ∀ vertex j∈Λ and τ∈[0,β], the total number of pairs (ux,i(τ+mβ),lx,i(τ+mβ)) with 0≤m<kx,i, and l(τ+mβ)=j does not exceed κ.Remark 18.
The integrals in (75) and (76) represent examples of partition functions which will be encountered in the forthcoming sections. See (96), (97), (101), (103), (105), (107), (111), and (113) below. A general form of such a partition function treated as an integral over a set of loop configurations rather than a trace in a Hilbert space is given in (96) and (97).


### 2.3. The Representation for the RDM Kernels

LetΛ0,Λ be finite sets, Λ0⊂Λ⊂Γ. The construction developed in Section 2.2 also allows us to write a convenient representation for the integral kernels of the RDMs RΛΛ0 (see (31)) and RΛ∣x¯Γ¯∖Λ*Λ0. In accordance with Lemma 11 and the definition of RΛΛ0 in (31), the operator RΛΛ0 acts as an integral operator in ℋ(Λ0):
(77)(RΛΛ0ϕ)(x*0)=∫M*Λ0∏j∈Λ0∏y∈y*(j)v(dy)FΛΛ0(x*0,y*0)ϕ(y*0),
where
(78)FΛΛ0(x*0,y*0):=∫M*Λ∖Λ0∏j∈Λ∖Λ0∏z∈z*(j)v(dz)×FΛ(x*0∨zΛ∖Λ0*,y*0∨zΛ∖Λ0*):=Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0)Ξ(Λ).We employ here and below the notationx*0 and y*0 for particle configurations xΛ0*={x*(i),i∈Λ0} and yΛ0*={y*(j),j∈Λ0} over Λ0. Next, x*0∨zΛ∖Λ0*; y*0∨zΛ∖Λ0* denotes the concatenated configurations over Λ.Similarly, the RDMRΛ∣x¯Γ¯∖Λ*Λ0 is determined by its integral kernel FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0), again admitting the representation
(79)FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0):=Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0∣x¯Γ¯∖Λ*)Ξ(Λ).As in [1], we call FΛΛ0 and FΛ∣x¯Γ¯∖Λ*Λ0 the RDM kernels (in short, RDMKs). The focus of our interest is the numerators Ξ^ΛΛ0(x*0,y*0,Λ∖Λ0) and Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0∣x¯Γ¯∖Λ*) in (78) and (79). To introduce the appropriate representation for these quantities, we need some additional definitions.Definition 19.
Repeating (57)-(58), symbol W¯x*0,y*0* denotes the disjoint union ⋃γ0‍W¯x*0,y*0,γ0* over matchings γ0 between x*0 and y*0. Accordingly, element Ω-*0=Ω-x*0,y*0,γ0*∈W¯x*0,y*0,γ0* yields a collection of paths Ω¯(x,i),γ0(x,i)*∈W¯(x,i),γ0(x,i)* lying in M×Γ. Each path Ω¯(x,i),γ0(x,i)* has time lengths βk(x,i),(y,j), begins at (x,i), and ends up at (y,j)=γ0(x,i), where x∈x*(i), y∈y*(j). Like above, we will use for Ω-*0 the term a path configuration over Λ0. Repeating (63)-(64), we obtain the measures ℙx*0,y*0,γ0* on W¯x*0,y*0,γ0* and ℙx*0,y*0* on W¯x*0,y*0*.The assertion of Lemma20 below again follows directly from known results, in conjunction with calculations of the partial trace trℋΛ∖Λ0 in ℋΛ. The meaning of new ingredients in (79)–(82) is explained below.Lemma 20.
The quantityΞ^ΛΛ0(x*0,y*0;Λ∖Λ0) emerging in (78) is set to be 0 when ♯x*0≠♯y*0. On the other hand, for ♯x*0=♯y*0,
(80)Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0)=∫W¯x*0,y*0*ℙ¯x*0,y*0*(dΩ-*0)B¯(Ω-*0)×αΛ(Ω-*0)1(Ω-*0∈ℱΛ0)×exp[-hΛ0(Ω-*0)]Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0),
where
(81)Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0)=∫WΛ∖Λ0*dΩΛ∖Λ0*B(ΩΛ∖Λ0*)×αΛ(ΩΛ∖Λ0*)1(ΩΛ∖Λ0*∈ℱΛ0)×exp[-hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0)].
Similarly, the quantity Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0∣x¯Γ¯∖Λ*) from (79) vanishes when ♯x*0≠♯y*0. For ♯x*0=♯y*0,
(82)Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0∣x¯Γ¯∖Λ*)=∫W¯x*0,y*0*ℙ¯x*0,y*0*(dΩ-*0)B¯(Ω-*0)×αΛ(Ω-*0)1(Ω-*0∈ℱΛ0)×exp[-hΛ0(Ω-*0∣x¯Γ¯∖Λ*)]×Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0∨x¯Γ¯∖Λ*),
where
(83)Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0∨x¯Γ¯∖Λ*)=∫WΛ∖Λ0*dΩΛ∖Λ0*B(ΩΛ∖Λ0*)αΛ(ΩΛ∖Λ0*)×1(ΩΛ∖Λ0*∈ℱΛ0)×exp[-hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0∨x¯Γ¯∖Λ*)].
These representations hold ∀z,β>0 and finite Λ0⊂Λ⊂Γ.Let us define the functionalsB¯(Ω-*0), αΛ(Ω-*0), 1(Ω-*0∈ℱΛ0), 1(ΩΛ∖Λ0*∈ℱΛ0)hΛ0(Ω-*0), hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0), hΛ0(Ω-*0x¯Γ¯∖Λ*), and hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0∨x¯Γ¯∖Λ*) in (79)–(83). (The functionals B(ΩΛ∖Λ0*) and αΛ(ΩΛ∖Λ0*) are defined as (70) and (74), respectively, replacing Λ with Λ∖Λ0.)To this end, letΩ-*0=Ω-x*0,y*0,γ0*∈W¯x*0,y*0,γ0* be a path configuration represented by a collection of paths Ω¯(x,i),γ0(x,i)*∈W¯(x,i),γ*0(x,i)* (Ω¯x,i* in short), with end points (x,i) and (y,j)=γ0(x,i), of time-length βk(x,i),(y,j). The functional B¯(Ω-*0) is given by
(84)B¯(Ω-*0)=∏i∈Λ‍∏x∈x*(i)‍zk(x,i),(y,j).
The functional αΛ(Ω-*0) is again an indicator:
(85)αΛ(Ω-*0)={1,ifΩ(x,i),γ0(x,i)*(τ)∈M×Λ,∀i∈Λ,x∈x*(i),0≤τ≤βk(x,i),(y,j),0,otherwise.Now let us define the indicator function1(·∈ℱΛ0) in (80)–(83). The factor 1(Ω-*0∈ℱΛ0) equals one if and only if every path Ω¯(x,i),(y,j)* from Ω-*0, of time-length βk(x,i),(y,j), starting at (x,i)∈M×Λ0 and ending up at (y,j)=γ0(x,i)∈M×Λ0 remains in M×(Λ∖Λ0) at the intermediate times βl for l=1,…,k(x,i),(y,j)-1:
(86)Ω¯(x,i),(y,j)*(lβ)∉M×Λ0,∀l=1,…,k(x,i),(y,j)-1,
(when  k(x,i),(y,j)=1, this is not a restriction).Furthermore, suppose thatΩΛ∖Λ0*=ΩxΛ∖Λ0** is a loop configuration over Λ∖Λ0, with the initial/end configuration xΛ∖Λ0*={x*(i),i∈Λ∖Λ0}, represented by a collection of loops Ωx,i*,i∈Λ∖Λ0,x∈x*(i). Then 1(ΩΛ∖Λ0*∈ℱΛ0)=1 if and only if each loop Ωx,i* of time-length βkx,i, beginning and finishing at (x,i)∈M×(Λ∖Λ0), does not enter the set M×Λ0 at times βl for l=1,…,kx,i-1:
(87)Ωx,i*(lβ)∉M×Λ0,∀l=1,…,kx,i-1
(again, if kx,i=1, this is not a restriction).The functionalhΛ0(Ω-*0) in (80) gives the energy of the path configuration Ω-*0 and is introduced similarly to (67), mutatis mutandis. Next, the functional hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0) in (81) represents the energy of the loop configuration ΩΛ∖Λ0* in the potential field generated by the path configuration Ω-*0:
(88)hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0)=hΛ∖Λ0(ΩΛ∖Λ0*)+h(Ω-*0∣∣ΩΛ∖Λ0*).
Here, the summand hΛ∖Λ0(ΩΛ∖Λ0*) yields the energy of the loop configuration ΩΛ∖Λ0*; again confer (67). Further, the term h(Ω-*0∣∣ΩΛ∖Λ0*) yields the energy of interaction between Ω-*0 and ΩΛ∖Λ0*: for a path/loop configurations Ω-*0={Ω¯x,i*}∈W¯x*0,y*0,γ0* and a ΩΛ∖Λ0*={Ωx′,i′*}∈WΛ∖Λ0* we set
(89)h(Ω-*0∣∣ΩΛ∖Λ0*)=∑(i,i′)∈Λ0×(Λ∖Λ0)∑x∈x*(i),x′∈x*(i′)h(x,i),(x′,i′)(Ω¯x,i*,Ωx′,i′*).
Here, for a path Ω¯x,i*=Ω¯(x,i),γ0(x,i)*, of time-length βk(x,i),γ0(x,i), and a loop Ωx′,i′*, of time-length βkx,i,
(90)h(x,i),(x′,i′)(Ω¯x,i*,Ωx′,i′*)=∑0≤m<k(x,i),γ0(x,i)∑0≤m′<kx′,i′∫0βdτ×[∑j∈ΓU(2)(u(τ+βm;Ω¯(x,i),γ0(x,i)*),u(τ+βm′;Ωx′,i′*))×1(l(x,i),γ0(x,i)(τ+βm)=j=lx′,i′(τ+βm′))+∑j,j′∈Γ×Γ‍J(d(j,j′))V(ui,x(τ+βm),ui′,x′(τ+βm′))×1(l(x,i),γ0(x,i)(τ+βm)=j≠j′=lx′,i′(τ+βm′))∑j∈ΓU(2)(u(τ+βm;Ω¯(x,i),γ0(x,i)*),].
Here, in turn, we employ the shortened notation for the positions and indices of the sections Ω¯(x,i),γ0(x,i)*(τ+βm) and Ωx′,i′*(τ+βm′) of Ω¯(x,i),γ0(x,i)* and Ωx′,i′* at times τ+βm and τ+βm′, respectively:
(91)ui,x(τ+βm)=u(τ+βm;Ω¯(x,i),γ0(x,i)*),li,x(τ+βm)=l(τ+βm;Ω¯(x,i),γ0(x,i)*),ui′,x′(τ+βm′)=u(τ+βm′;Ωx′,i′*),li′,x′(τ+βm′)=l(τ+βm′;Ωx′,i′*).Further, the functionalhΛ0(Ω-*0∣x¯Γ¯∖Λ*) in (82) is determined as in (71)–(73), with Ω¯(x,i),γ0(x,i)* instead of Ωx,i*. Next, for hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0∨x¯Γ¯∖Λ*) in (83), we set
(92)hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0∨x¯Γ¯∖Λ*)=hΛ∖Λ0(ΩΛ∖Λ0*)+h(ΩΛ∖Λ0*∣∣Ω-*0∨x¯Γ¯∖Λ*).
Here again, the summand hΛ∖Λ0(ΩΛ∖Λ0*) is determined as in (67). Next, the term hΛ∖Λ0(ΩΛ∖Λ0*∣∣Ω-*0∨x¯Γ¯∖Λ*) is defined similarly to (72)-(73):
(93)hΛ∖Λ0(ΩΛ∖Λ0*∣∣Ω-*0∨x¯Γ¯∖Λ*)=∑(i,i′)∈(Λ∖Λ0)×Λ0∑x∈x*(i),x′∈x*(i′)h(x,i),(x′,i′)(Ωx,i*,Ω¯x′,i′*)+∑i∈Λ∖Λ0∑x∈x*(i),x-′∈x¯*(i′)h(x,i),(x-′,i′)(Ωx,i*,(x-′,i′))
with
(94)h(x,i),(x′,i′)(Ωx,i*,Ω¯x′,i′*)=∑0≤m<kx,i∑0≤m′<k(x′,i′),γ0(x′,i′)×∫0βdτ×[∑j∈Γ‍U(2)(ux,i(τ+βm),ux′,i′(τ+βm′))×1(lx,i(τ+βm)=j=lx′,i′(τ+βm′))+∑(j,j′)∈Γ×ΓJ(d(j,j′))V×(ui,x(τ+βm),ux′,i′(τ+βm′))×1(lx,i(τ+βm)=j≠j′=lx′,i′(τ+βm′))∑j∈Γ‍],h(x,i),(x-′,i′)(Ωx,i*,(x-′,i′))=∑0≤m<kx,i∫0βdτ[(ui,x(τ+βm),x-′)1(lx,i(τ+βm)=j)U(2)(ux,i(τ+βm),x-′)×1(lx,i(τ+βm)=i′)+∑j∈Γ‍J(d(j,i′))V(ui,x(τ+βm),x-′)×1(lx,i(τ+βm)=j)(ui,x(τ+βm),x-′)1(lx,i(τ+βm)=j)U(2)].As before, the functionalshΛ(ΩΛ*) and hΛ(ΩΛ*∣∣x¯Γ¯∖Λ*) have a natural interpretation as energies of loop configurations.Repeating the above observation, non-zero contributions to the integral in (80) come only from pairs (Ω-*0,ΩΛ∖Λ0*) such that ∀j∈Γ and τ∈[0,β], the total number of pairs (u(τ+βm;Ω¯(x,i),γ0(x,i)*),l(τ+βm;Ω¯(x,i),γ0(x,i)*)) with 0≤m<k(x,i),γ0(x,i), i∈Λ0 and x∈x*(i) incident to the paths of the configuration Ω-*0 and pairs (u(τ+βm′;Ωx′,i′*),l(τ+βm′;Ωx′,i′*)) with 0≤m′<k(x′,i′), i′∈Λ∖Λ0 and x′∈x*(i′) incident to the loops of the configuration Ω-Λ∖Λ0* does not exceed κ. Similarly, non-zero contributions to the integral in (82) come only from pairs (Ω-*0,ΩΛ∖Λ0*) such that the above inequality holds when we additionally count points x¯∈x¯*(j).The integralΞ^ΛΛ0(Λ∖Λ0∣Ω-*0) defined in (81) can be considered as a particular (although important) example of a partition function in the volume Λ∖Λ0 with a boundary condition Ω-*0. Note the presence of the subscript Λ indicating that the loops contributing to Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0) can jump within volume Λ only (owing to the indicator functional αΛ). On the other hand, the presence of the indicator functional 1(ΩΛ∖Λ0*∈ℱΛ0) in the integral (reflected in the upperscript Λ0 and the roof sign in the notation Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0)) indicates a particular restriction on the jumps of the loops, forbidding them to visit set Λ0 at intermediate times βm. This is true also for the integral Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0∨x¯Γ¯∖Λ*) in (83): it is a particular example of a partition function in the volume Λ∖Λ0 with a boundary condition Ω-*0∨x¯Γ¯∖Λ*.Other useful types of partition functions areΞΓ0(Λ~∣Ω-*0∨ΩΓ1*) andΞΓ0(Λ~∣Ω-*0∨ΩΓ1*∨x¯Γ2*) where the sets of vertices Λ~, Λ0, Γ0, Γ1, and Γ2 satisfy
(95)Λ~⊂Γ0⊆Γ,Γ1,Γ2⊂Γ∖Λ~,Γ1∩Γ2=∅,Λ0⊂Γ∖(Λ~∪Γ1∪Γ2)
and ♯Λ~,♯Λ0<+∞. Accordingly, Ω-*0 is a (finite) configuration over Λ0, ΩΓ1* a (possibly infinite) loop configuration over Γ1, and x¯Γ2* a (possibly infinite) particle configuration over Γ2. The partition functions ΞΓ0(Λ~∣Ω-*0∨ΩΓ1*) and ΞΓ0(Λ~∣Ω-*0∨ΩΓ1*∨x¯Γ2*) are given by
(96)ΞΓ0(Λ~∣Ω-*0∨ΩΓ1*)=∫WΛ~*dΩΛ~*αΓ0(ΩΛ~*)B(ΩΛ~*)×exp[-hΛ~(ΩΛ~*∣Ω-*0∨ΩΓ1*)],(97)ΞΓ0(Λ~∣Ω-*0∨ΩΓ1*∨x¯Γ2*)=∫WΛ~*dΩΛ~*αΓ0(ΩΛ~*)B(ΩΛ~*)×exp[-hΛ~(ΩΛ~*∣Ω-*0∨ΩΓ1*∨x¯Γ2*)]
with the indicator αΓ0 as in (74). These partition functions, feature loop configurations Ωλ~* formed by loops Ωx,i*, i∈Λ~, which start and finish in Λ~, are confined to Γ0 and move in a potential field generated by Ω-*0∨ΩΓ1*, where Ω-*0={Ω¯(x,i),γ0(x,i)*} and ΩΓ1={Ωx,i*,x∈x*(i),i∈Γ1} or ΩΓ1*∨x¯Γ2*, where x¯Γ2*={x¯*(i),i∈Γ2}. (The latter can be understood as the concatenation of the loop configuration ΩΓ1* over Γ1 and the loop configuration over Γ2 formed by the constant trajectories sitting at points x¯∈x¯*(i), i∈Γ2.) In (96) we assume that, ∀τ∈[0,β] and j∈Γ, the number
(98)♯{(x,i,m):i∈Λ~,l(τ+mβ;Ωx,i*)=j,0≤m<kx,i}+♯{(x,i):i∈Λ0,l(τ+mβ;Ω¯(x,i),γ0(x,i)*)=j,0≤m<k(x,i),γ0(x,i)}+♯{(x,i,m):i∈Γ1,l(τ+mβ;Ωx,i*)=j,0≤m<kx,i}
does not exceed κ. Analogously, in (97) it is assumed that the same is true for the above number plus the cardinality ♯x¯*(j).Such “modified” partition functions will be used in forthcoming sections.


### 2.4.  The FK-DLR MeasureμΛ in a Finite Volume

The Gibbs statesφΛ and φΛ∣x¯Γ¯∖Λ* give rise to probability measures μΛ and μΛ∣x¯Γ¯∖Λ* on the sigma algebra WΛ of subsets of WΛ*. The sigma algebra WΛ is constructed by following the structure of the space WΛ* (a disjoint union of Cartesian products); confer Definition 16. The measures μΛ and μΛ∣x¯Γ¯∖Λ* are determined by their Radon-Nykodym derivative pΛ and pΛ∣x¯Γ¯∖Λ* relative to the measure dΩΛ*:
(99)pΛ(ΩΛ*)∶=μΛ(dΩΛ*)dΩΛ*=1Ξ(Λ)αΛ(ΩΛ*)B(ΩΛ*)×exp[-hΛ(ΩΛ*)],ΩΛ*∈WΛ*,pΛ∣x¯Γ¯∖Λ*(ΩΛ*)∶=μΛ∣x¯Γ¯∖Λ*(dΩΛ*)dΩΛ*1Ξ(Λ∣x¯Γ¯∖Λ*)=αΛ(ΩΛ*)B(ΩΛ*)×exp[-hΛ(ΩΛ*∣x¯Γ¯∖Λ*)],ΩΛ*∈WΛ*.
Given Λ0⊂Λ, the sigma algebra WΛ0 is naturally identified with a sigma subalgebra of WΛ. The restrictions of μΛ to WΛ0 and μΛ∣x¯Γ¯∖Λ* are denoted by μΛΛ0 and μΛ∣x¯Γ¯∖Λ*Λ0; these measures are determined by their Radon-Nikodym derivatives pΛΛ0(ΩΛ0*):=μΛΛ0(dΩΛ0*)/dΩΛ0* and pΛ∣x¯Γ¯∖Λ*Λ0(ΩΛ0*):=μΛ∣x¯Γ¯∖Λ*Λ0(dΩΛ0*)/dΩΛ0*.The first key property of the measuresμΛ and μΛ∣x¯Γ¯∖Λ* is expressed in the so-called FK-DLR equation. We state it as Lemma 21 below; its proof repeats a standard argument used in the classical case for establishing the DLR equation in a finite volume Λ⊂Γ.Lemma 21.
For allz,β>0 satisfying (22), and Λ0⊂Λ′⊂Λ, the probability density pΛΛ0 admits the form:
(100)pΛΛ0(ΩΛ0*)=∫WΛ∖Λ′*qΛ∖Λ′Λ0(ΩΛ0*∣ΩΛ∖Λ′*)μΛΛ∖Λ′(dΩΛ∖Λ′*),
where
(101)qΛ∖Λ′Λ0(ΩΛ0*∣ΩΛ∖Λ′*)=exp[-hΛ0(ΩΛ0*∣ΩΛ∖Λ′*)]×ΞΛ(Λ′∖Λ0∣ΩΛ0*∨ΩΛ∖Λ′*)ΞΛ(Λ′∣ΩΛ∖Λ′*),
and the conditional partition functions ΞΛ(Λ′∖Λ0∣Ω*∨ΩΛ∖Λ′*) andΞΛ(Λ′∣ΩΛ∖Λ′*) are determined as in (96).
Similarly, forpΛ∣x¯Γ¯∖Λ*Λ0 one has:
(102)pΛ∣x¯Γ¯∖Λ*Λ0(ΩΛ0*)=∫WΛ∖Λ′*qΛ∖Λ′Λ0(ΩΛ0*∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*)×μΛ∣x¯Γ¯∖Λ*Λ∖Λ′(dΩΛ∖Λ′*),
where
(103)qΛ∖Λ′Λ0(ΩΛ0*∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*)=exp[-hΛ0(ΩΛ0*∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*)]×ΞΛ(Λ′∖Λ0∣ΩΛ0*∨ΩΛ∖Λ′*∨x¯Γ¯∖Λ*)ΞΛ(Λ′∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*)
and the conditional partition functions ΞΛ(Λ′∖Λ0∣ΩΛ0*∨ΩΛ∖Λ′*∨x¯Γ¯∖Λ*) andΞΛ(Λ′∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*) are determined as in (96).As in [1], (100) and (102) mean that the conditional densities pΛΛ0(ΩΛ0*∣ΩΛ∖Λ′*) and pΛ∣x¯Γ¯∖Λ*Λ0(ΩΛ0*∣ΩΛ∖Λ′*) relative to σ-algebra WΛ∖Λ′ coincide, respectively, with qΛ∖Λ′Λ0(Ω*0∣ΩΛ∖Λ′*), and qΛ∖Λ′Λ0(Ω*0∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*), for μΛΛ∖Λ′—and μΛ∣x¯Γ¯∖Λ*Λ∖Λ′—a.a. ΩΛ∖Λ′*∈WΛ∖Λ′* and a.a. ΩΛ0*∈WΛ0*.As in [1], we call the expressions qΛ∖Λ′Λ0(ΩΛ0*∣ΩΛ∖Λ′*) and qΛ∖Λ′Λ0(ΩΛ0*∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*), as well as the expressions q^Λ∖Λ′Λ0(Ω-*0∣ΩΛ∖Λ′*) and q^Λ∖Λ′Λ0(Ω-*0∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*) appearing below, the (conditional) RDM functionals (in brief, the RDMFs). The same name will be used for the quantity qΛ∖Λ′Λ0(ΩΛ0*∣ΩΛ∖Λ′*) from (110)-(111) and the quantity q^Γ∖ΛΛ0(Ω-*0∣ΩΓ∖Λ*) from (112)-(113).The second property is that the RDMKsFΛΛ0(x*0,y*0) and FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0) are related to the measures μΛ and μΛ∣x¯Γ¯∖Λ′*. Again, the proof of this fact is done by inspection.Lemma 22.
The RDMKFΛΛ0(x*0,y*0) is expressed as follows: ∀Λ0⊂Λ′⊂Λ,
(104)FΛΛ0(x*0,y*0)=∫W¯x0,y0*ℙ¯x0,y0*(dΩ-*0)αΛ(Ω-*0)×B¯(Ω-*0)1(Ω-*0∈ℱΛ0)×∫WΛ∖Λ′*μΛ∖Λ′(dΩΛ∖Λ′*)1(ΩΛ∖Λ′*∈ℱΛ0)×q^Λ∖Λ′Λ0(Ω-*0∣ΩΛ∖Λ′*),
where
(105)q^Λ∖Λ′Λ0(Ω-*0∣ΩΛ∖Λ′*)=exp[-hΛ0(Ω-*0∣ΩΛ∖Λ′*)]×Ξ^ΛΛ0(Λ′∖Λ0∣Ω-*0∨ΩΛ∖Λ′*)ΞΛ(Λ′∣ΩΛ∖Λ′*).
Similarly,(106)FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0)=∫W¯x0,y0*ℙ¯x0,y0*(dΩ-*0)αΛ(Ω-*0)×B¯(Ω-*0)1(Ω-*0∈ℱΛ0)×∫WΛ∖Λ′*μΛ∣x¯Γ¯∖Λ*Λ∖Λ′(dΩΛ∖Λ′*)1(ΩΛ∖Λ′*∈ℱΛ0)×q^Λ∖Λ′Λ0(Ω-*0∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*),
where
(107)q^Λ∖Λ′Λ0(Ω-*0∣ΩΛ∖Λ′*x¯Γ¯∖Λ*)=exp[-hΛ0(Ω-*0∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*)]×Ξ^ΛΛ0(Λ′∖Λ0∣Ω-*0∨ΩΛ∖Λ′*∨x¯Γ¯∖Λ*)ΞΛ(Λ′∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*).
Here the partition functions Ξ^ΛΛ0(Λ′∖Λ0∣Ω-*0∨ΩΛ∖Λ′*) and Ξ^ΛΛ0(Λ′∖Λ0∣Ω-*0∨ΩΛ∖Λ′*∨x¯Γ¯∖Λ*) are determined in (81) and (83).Remark 23.
Summarizing the above observations, the measuresμΛ and μΛ∣x¯Γ-∖Λ* are concentrated on the subset in WΛ* formed by loop configurations ΩΛ* such that ∀τ∈[0,β], the section ΩΛ*(τ) has ≤κ particles at each vertex i∈Λ.


## 2.1.  The Representation for the Kernels of the Gibbs Operators

A starting point for the forthcoming analysis is the Feynman-Kac (FK) representation for the kernelsKΛ(xΛ*,yΛ*)=Kβ,z,Λ(xΛ*,yΛ*) and FΛ(xΛ*,yΛ*)=Fβ,z,Λ(xΛ*,yΛ*) of operators GΛ and RΛ.Definition 7.
Given(x,i),(y,j)∈M×Γ, W¯(x,i),(y,j)β denotes the space of path, or trajectories, ω¯=ω¯(x,i),(y,j) in M×Γ, of time-length β, with the end-points (x,i) and (y,j). Formally, ω¯∈W¯(x,i),(y,j)β is defined as follows:
(37)ω¯:τ∈[0,β]⟼ω¯(τ)=(u(ω¯,τ),l(ω¯,τ))∈M×Γ,ω¯isca´dla´g;ω¯(0)=(x,i),ω¯(β-)=(y,j),ω¯hasfinitelymanyjumpson[0,β];ifajumpoccursattimeτ,thend[l(ω-,τ-),l(ω-,τ)]=1.
The notation ω¯(τ) and its alternative, (u(ω¯,τ),l(ω¯,τ)), for the position and the index of trajectory ω¯ at time τ will be employed as equal in rights. We use the term the temporal section (or simply the section) of path ω¯ at time τ.Definition 8.
LetxΛ*={x*(i),i∈Λ}∈M*Λ, and yΛ*={y*(j),j∈Λ}∈M*Λ be particle configurations over Λ, with ♯xΛ*=♯yΛ*. A matching (or pairing) γ between xΛ* and yΛ* is defined as a collection of pairs [(x,i),(y,j)]γ, with i,j∈Λ, x∈x*(i), and y∈y*(j), with the properties that (i) ∀i∈Λ and x∈x*(i): there exist unique j∈Λ and y∈y*(j) such that (x,i) and (y,j) form a pair, and (ii) ∀j∈Λ and y∈y*(j): there exist unique i∈Λ and x∈x*(i) such that (x,i) and (y,j) form a pair. (Owing to the condition ♯xΛ*=♯yΛ*, these properties are equivalent.) It is convenient to write [(x,i),(y,j)]γ=[(x,i),γ(x,i)].
Next, consider the Cartesian product(38)W¯xΛ*,yΛ*,γβ=×i∈Λ×x∈x*(i)W¯(x,i),γ(x,i)β,
and the disjoint union
(39)W¯xΛ*,yΛ*β=⋃‍γW¯xΛ*,yΛ*,γβ.
Accordingly, an element ω¯Λ∈W¯xΛ*,yΛ*,γβ in (38) represents a collection of paths ω¯x,i, x∈x*(i), i∈Λ, of time-length β, starting at (x,i) and ending up at γ(x,i). We say that ω¯Λ is a path configuration in (or over) Λ.The presence of matchings in the above construction is a feature of the bosonic nature of the systems under consideration.We will work with standard sigma algebras (generated by cylinder sets) inW¯(x,i),(y,j)β, W¯xΛ*,yΛ*,γβ, and W¯xΛ*,yΛ*β.Definition 9.
In what follows,ξ(τ), τ≥0, stands for the Markov process on M×Γ, with cádlág trajectories, determined by the generator ℒ acting on a function (x,i)∈M×Λ↦ψ(x,i) by
(40)ℒψ(x,i)=-12Δψ(x,i)ℒψ(x,i)=+∑j:d(i,j)=1λi,j∫Mv(dy)[ψ(y,j)-ψ(x,i)].
In the probabilistic literature, such processes are referred to as Lévy processes; see, for example, [14].Pictorially, a trajectory of processξ moves along M according to the Brownian motion with the generator -Δ/2 and changes the index i∈Γ from time to time in accordance with jumps occurring in a Poisson process of rate ∑j:d(i,j)=1λi.j. In other words, while following a Brownian motion rule on M, having index i∈Γ and being at point x∈M, the moving particle experiences an urge to jump from i to a neighboring vertex j and to a point y at rate λi,jv(dy). After a jump, the particle continues the Brownian motion on M from y and keeps its new index j until the next jump, and so on.For a given pairs(x,i),(y,j)∈M×Γ, we denote by ℙ(x,i),(y,j)β the nonnormalised measure on W¯(x,i),(y,j)β induced by ξ. That is, under measure ℙ(x,i),(y,j)β the trajectory at time τ=0 starts from the point x and has the initial index i while at time τ=β it is at the point y and has the index j. The value p^(x,i),(y,j)=ℙ(x,i),(y,j)β(W¯(x,i),(y,j)β) is given by
(41)p^(x,i),(y,j)=1(i=j)pMβ(x,y)exp[-β∑j~:d(i,j~)=1λi,j~]+∑k≥1∑l0=i,l1,…,lk,lk+1=j∏0≤s≤k1(d(ls,ls+1)=1)×λls,ls+1∫0βdτsexp[-(τs+1-τs)∑j~:d(ls,j~)=1λls,j~]×1(0=τ0<τ1<⋯<τk<τk+1=β),
where pMβ(x,y) denotes the transition probability density for the Brownian motion to pass from x to y on M in time β:
(42)pMβ(x,y)=1(2πβ)d/2×∑n_=(n1,…,nd)∈ℤdexp(-|x-y+n_|22β).
In view of (13), the quantity p^(x,i),(y,j) and its derivatives are uniformly bounded:
(43)p^(x,i),(y,j),|∇xp^(x,i),(y,j)|,|∇yp^(x,i),(y,j)|≤p^M,p^(x,i),(y,j),|∇xp^(x,i),(y,j)|1,x,y∈M,i,j∈Γ,
where p^M=p^M(β)∈(0,+∞) is a constant.We suggest a term “non-normalised Brownian bridge with jumps” for the measure but expect that a better term will be proposed in future.Definition 10.
Suppose thatxΛ*={x*(i),i∈Λ}∈M*Λ and yΛ*={y*(j),j∈Λ}∈M*Λ are particle configurations over Λ, with ♯xΛ*=♯yΛ*. Let γ be a pairing between xΛ* and yΛ*. Then ℙxΛ*,yΛ*,γ* denotes the product measure on W¯xΛ*,yΛ*,γβ:
(44)ℙ¯xΛ*,yΛ*,γβ=×i∈Λ×x∈x*(i)ℙ¯(x,i),γ(x,i)β.
Furthermore, ℙ¯xΛ*,yΛ*β stands for the sum measure on W¯xΛ*,yΛ*β:
(45)ℙ¯xΛ*,yΛ*β=∑γ‍ℙ¯xΛ*,yΛ*,γβ.According to Definition10, under the measure ℙ¯xΛ*,yΛ*,γβ, the trajectories ω¯x,i∈W¯(x,i),γ(x,i)β constituting ω¯Λ are independent components. (Here the term independence is used in the measure-theoretical sense.)As in [1], we will work with functionals on W¯xΛ*,yΛ*,γβ representing integrals along trajectories. The first such functional, hΛ(ω¯Λ), is given by
(46)hΛ(ω¯Λ)=∑i∈Λ∑x∈x*(i)hx,i(ω¯x,i)hΛ(ω¯Λ)=+12∑(i,i′)∈Λ×Λ∑x∈x*(i),x′∈x*(i′)h(x,i),(x′,i′)hΛ(ω¯Λ)=×(ω¯x,i,ω¯x′,i′).
Here, introducing the notation ux,i(τ)=u(ω¯x,i,τ) and ux′,i′(τ)=u(ω¯x′,i′,τ) for the positions in M of paths ω¯x,i∈W¯(x,i),γ(x,i)β and ω¯x′,i′∈W¯(x′,i′),γ(x′,i′)β at time τ, we define
(47)hx,i(ω¯x,i)=∫0βdτU(1)(ui,x(τ)).
Next, with lx,i(τ) and lx′,i′(τ) standing for the indices of ω¯x,i and ω¯x′,i′ at time τ,
(48)h(x,i),(x′,i′)(ω¯x,i,ω¯x′,i′)=∫0βdτ[∑j′∈ΓU(2)(ui,x(τ),ui′,x′(τ))×1(lx,i(τ)=j′=lx′,i′(τ))+12∑(j′,j′′)∈Γ×ΓJ(d(j′,j′′))×V(ui,x(τ),ui′,x′(τ))×1(lx,i(τ)=j′≠j′′=lx′,i′(τ))∑j′∈ΓU(2)(ui,x(τ),ux′,i′(τ))].Next, consider the functionalhΛ(ω¯Λ∣x¯Γ¯∖Λ*): for x¯Γ¯∖Λ*={x¯*(j),j∈Γ¯∖Λ}. As before, we assume that ♯x¯*(j)≤κ. Define
(49)hΛ(ω¯Λ∣x¯Γ¯∖Λ*)=hΛ(ω¯Λ)+hΛ(ω¯Λ∣∣x¯Γ¯∖Λ*).
Here hΛ(ω¯Λ) is as in (46) and
(50)hΛ(ω¯Λ∣∣x¯Γ¯∖Λ*)=∑(i,i′)∈Λ×(Γ¯∖Λ)∑x∈x*(i),x-′∈x¯*(i′)h(x,i),(x-′,i′)×(ω¯x,i,(x-′,i′)),
where, in turn,
(51)h(x,i),(x-′,i′)(ω¯x,i,(x-′,i′))=∫0βdτ[U(2)(ui,x(τ),x-′)1(lx,i(τ)=i′)+∑j∈Γ:j≠i′J(d(j,i′))×V(ui,x(τ),x-′)1(lx,i(τ)=j)].
The functionals hΛ(ω¯Λ) and hΛ(ω¯Λ∣∣x¯Γ¯∖Λ*) are interpreted as energies of path configurations. Compare (2.1.4) and (2.3.8) in [1].Finally, we introduce the indicator functionalαΛ(ω¯Λ):
(52)αΛ(ω¯Λ)={1,ifindexlx,i(τ)∈Λ,∀τ∈[0,β],i∈Λ,x∈x*(i),0,otherwise.
It can be derived from known results [11, 15–17] (for a direct argument, see [18]) that the following assertion holds true.Lemma 11.
For allz,β>0 and a finite Λ, the Gibbs operators GΛ and GΛ∣x¯Γ¯∖Λ* act as integral operators in ℋ(Λ):
(53)(GΛϕ)(xΛ*)=∫M*Λ∏j∈Λ∏y∈y*(j)v(dy)KΛ(xΛ*,yΛ*)ϕ(yΛ*),(GΛ∣x¯Γ¯∖Λ*ϕ)(xΛ*)=∫M*Λ∏j∈Λ∏y∈y*(j)v(dy)KΛ(xΛ*,yΛ*∣x¯Γ¯∖Λ*)ϕ(yΛ*).
Moreover, the integral kernels KΛ(xΛ*,yΛ*) and KΛ(xΛ*,yΛ*∣x¯Γ¯∖Λ*) vanish if ♯xΛ*≠♯yΛ*. On the other hand, when ♯xΛ*=♯yΛ*, the kernels KΛ(xΛ*,yΛ*) and KΛ(xΛ*,yΛ*∣x¯Γ¯∖Λ*) admit the following representations:
(54)KΛ(xΛ*,yΛ*)=z♯xΛ*∫W¯xΛ*,yΛ**ℙ¯xΛ*,yΛ*β(dω¯Λ)αΛ(ω¯Λ)×exp[-hΛ(ω¯Λ)],(55)KΛ(xΛ*,yΛ*∣x¯Γ¯∖Λ*)=z♯xΛ*∫W¯xΛ*,yΛ**ℙ¯xΛ*,yΛ*β(dω¯Λ)αΛ(ω¯Λ)×exp[-hΛ(ω¯Λ∣x¯Γ¯∖Λ*)].
The ingredients of these representations are determined in (46)–(51).Remark 12.
As before, we stress that, owing to (16) and (17), a nonzero contribution to the integral in the RHS of (54) can only come from a path configuration ω¯Λ={ω¯x,i} such that ∀τ∈[0,β] and ∀j∈Γ, the number of paths ω¯x,i with index lx,i(τ)=j is less than or equal to κ. Likewise, the integral in the RHS of (55) receives a non-zero contribution only from configurations ω¯Λ={ω¯x,i*} such that, ∀ site j∈Γ, the number of paths ω¯x,i with index lx,i(τ)=j plus the cardinality ♯x¯*(j) does not exceed κ.


## 2.2. The Representation for the Partition Function

The FK representations of the partition functionsΞ(Λ)=Ξβ,z(Λ) in (27) and Ξ(Λ∣x¯Γ¯∖Λ*) in (1.4.6) reflect a specific character of the traces trGΛ and trGΛ∣x¯Γ¯∖Λ* in ℋ(Λ). The source of a complication here is the jump terms in the Hamiltonians HΛ and HΛ∣x¯Γ¯∖Λ* in (11) and (12), respectively. In particular, we will have to pass from trajectories of fixed time-length β to loops of a variable time length. To this end, a given matching γ is decomposed into a product of cycles, and the trajectories associated with a given cycle are merged into closed paths (loops) of a time-length multiple of β. (A similar construction has been performed in [18].)To simplify the notation, we omit, wherever possible, the indexβ.Definition 13.
For given(x,i),(y,j)∈M×Γ, the symbol W¯(x,i),(y,j)* denotes the disjoint union:
(56)W¯(x,i),(y,j)*=⋃k=0,1,…W¯(x,i),(y,j)kβ.
In other words, W¯(x,i),(y,j)* is the space of paths Ω¯*=Ω¯(x,i),(y,j)* in M×Γ, of a variable time-length kβ, where k=k(Ω¯*) takes values 1,2,… and called the length multiplicity, with the end-points (x,i) and (y,j). The formal definition follows the same line as in (37), and we again use the notation Ω¯*(τ) and the notation (u(Ω¯*,τ),l(Ω¯*,τ)) for the pair of the position and the index of path Ω¯* at time τ. Next, we call the particle configuration {Ω¯*(τ+βm),0≤m<k(Ω¯*)} the temporal section (or simply the section) of Ω¯* at time τ∈[0,β]. We also call Ω¯(x,i),(y,j)*∈W¯(x,i),(y,j)* a path (from (x,i) to (y,j)).
A particular role will be played by closed paths (loops), with coinciding endpoints (where(x,i)=(y,j)). Accordingly, we denote by Wx,i* the set W¯(x,i),(x,i)*. An element of Wx,i* is denoted by Ωx,i* or, in short, by Ω* and called a loop at vertex i. (The upper index * indicates that the length multiplicity is unrestricted.) The length multiplicity of a loop Ωx,i*∈Wx,i* is denoted by k(Ωx,i*) or kx,i. It is instructive to note that, as topological object, a given loop Ω* admits a multiple choice of the initial pair (x,i): it can be represented by any pair (u(Ω*,τ),l(Ω*,τ)) at a time τ=lβ where l=1,…,k(Ω*). As above, we use the term the temporal section at time τ∈[0,β] for the particle configuration {Ωx,i*(τ+βm),0≤m<kx,i} and employ the alternative notation (u(τ+βm;Ω*),l(τ+βm;Ω*)) addressing the position and the index of Ω* at time τ+βm∈[0,βk(Ω*)].Definition 14.
SupposexΛ*={x*(i),i∈Λ}∈M*Λ and yΛ*={y*(j),j∈Λ}∈M*Λ are particle configurations over Λ, with ♯xΛ*=♯yΛ*. Let γ be a matching between xΛ* and yΛ*. We consider the Cartesian product:
(57)W¯xΛ*,yΛ*,γ*=×i∈Λ×x∈x*(i)W¯(x,i),γ(x,i)*,
and the disjoint union:
(58)W¯xΛ*,yΛ**=⋃γW¯xΛ*,yΛ*,γ*.
Accordingly, an element Ω-Λ*∈W¯xΛ*,yΛ*,γ* in (58) represents a collection of paths Ω-x,i*, x∈x*(i), i∈Λ, of time-length kβ, starting at (x,i) and ending up at (y,j)=γ(x,i). We say that Ω-Λ*∈W¯xΛ*,yΛ** is a path configuration in (or over) Λ.
Again, loops play a special role and deserve a particular notation. Namely,WxΛ** denotes the Cartesian product:
(59)WxΛ**=×i∈Λ×x∈x*(i)Wx,i*,
and WΛ* stands for the disjoint union (or equivalently, the Cartesian power):
(60)WΛ*=⋃xΛ*∈M*ΛWxΛ**=×i∈ΛW{i}*,whereW{i}*=⋃x*∈M*(×x∈x*Wx,i*).
Denote byΩ*={Ω*(i),i∈Λ}∈WΛ* a collection of loop configurations at vertices i∈Λ starting and ending up at particle configurations x*(i)∈M* (note that some of the Ω*(i)’s may be empty). The temporal section (or, in short, the section), Ω*(τ), of Ω* at time τ is defined as the particle configuration formed by the points Ωx,i*(τ+βm) where i∈Λ, x∈x*(i), and 0≤m<kx,i.As before, consider the standard sigma algebras of subsets in the spacesW¯(x,i),(y,j)*, Wx,i, W¯xΛ*,yΛ*,γ*, W¯xΛ*,yΛ**, WxΛ**, and WΛ* introduced in Definitions 13 and 14. In particular, the sigma algebra of subsets in WΛ* will be denoted by WΛ; we comment on some of its specific properties in Section 3.1. (An infinite-volume version WΓ* of WΛ* is treated in Section 3.2 and after.)Definition 15.
Given points(x,i),(y,j)∈M×Γ, we denote by ℙ¯(x,i),(y,j)* the sum measure on W¯(x,i),(y,j)*:
(61)ℙ¯(x,i),(y,j)*=∑k=0,1,…ℙ¯(x,i),(y,j)kβ.
Further,ℙx,i* denotes the similar measure on Wx,i*:
(62)ℙx,i*=∑k=0,1,…ℙx,ikβ.Definition 16.
LetxΛ*={x*(i),i∈Λ}∈M*Λ and yΛ*={y*(j),j∈Λ}∈M*Λ be particle configurations over Λ, with ♯xΛ*=♯yΛ*. Let γ be a matching between xΛ* and yΛ*, and we define the product measure ℙ¯xΛ*,yΛ*,γ*:
(63)ℙ¯xΛ*,yΛ*,γ*=×i∈Λ×x∈x*(i)ℙ¯(x,i),γ(x,i)*
and the sum measure
(64)ℙ¯xΛ*,yΛ**=∑γℙ¯xΛ*,yΛ*,γ*.
Next, symbolℙxΛ** stands for the product measure on WxΛ**:
(65)ℙxΛ**=×i∈Λ×x∈x*(i)ℙx,i*.
Finally,dΩΛ* yields the measure on WΛ*:
(66)dΩΛ*=dxΛ*×ℙxΛ**(dΩΛ*).
Here, for xΛ*={x*(i),i∈Λ}, we set: dxΛ*=∏i∈Λ∏x∈x*(i)v(dx). For sites i with x*(i)=∅, the corresponding factors are trivial measures sitting on the empty configurations.We again need to introduce energy-type functionals represented by integrals along loops. More precisely, we define the functionalshΛ(ΩΛ*) and hΛ(ΩΛ*∣x¯Γ¯∖Λ*) which are modifications of the above functionals hΛ(ω¯Λ) and hΛ(ω¯Λ∣x¯Γ¯∖Λ*); confer (46) and (50). Say, for a loop configuration ΩΛ*={Ωx,i*} over Λ with an initial and final particle configuration xΛ*={x*(i),i∈Λ},
(67)hΛ(ΩΛ*)=∑i∈Λ∑x∈x*(i)hx,i(Ωx,i*)+12∑(i,i′)∈Λ×Λ∑x∈x*(i),x′∈x*(i′)×1((x,i)≠(x′,i′))h(x,i),(x′,i′)(Ωx,i*,Ωx′,i′*).
To determine the functionals hx,i(Ωx,i*) and h(x,i),(x′,i′)(Ωx,i*,Ωx′,i′*), we set, for given m=0,1,…,kx,i-1 and m′=0,1,…,kx′,i′-1:
(68)ui,x(τ+βm)=u(τ+βm;Ωx,i*),li,x(τ+βm)=l(τ+βm;Ωx,i*),ui′,x′(τ+βm′)=u(τ+βm′;Ωx′,i′*),li′,x′(τ+βm′)=l(τ+βm′;Ωx′,i′*).
A (slightly) shortened notation li,x(τ+βm) is used for the index lx,i(τ+βm;Ωx,i*) and ui,x(τ+βm) for the position u(τ+βm;Ωx,i*) for Ωx,i*(τ)∈M×Γ, of the section Ωx,i*(τ) of the loop Ωx,i* at time τ, and similarly with li′,x′(τ+βm′) and ui′,x′(τ+βm′). (Note that the pairs (x,i) and (x′,i′) may coincide.) Then
(69)h(x,i)(Ωx,i*)=∫0βdτ[∑0≤m<kx,iU(1)(ux,i(τ+βm))+∑0≤m<m′<kx,i∑j∈Γ1(lx,i(τ+βm)=j=lx,i(τ+βm′))×U(2)(ux,i(τ+βm),ux,i(τ+βm′))∑0≤m<kx,iU(1)(ux,i(τ+βm))],h(x,i),(x′,i′)(Ωx,i*,Ωx′,i′*)=∑0≤m<kx,i∑0≤m′<kx′,i′∫0βdτ×[∑j∈ΓU(2)(ux,i(τ+βm),ux′,i′(τ+βm′))×1(lx,i(τ+βm)=j=lx′,i′(τ+βm′))+∑(j,j′)∈Γ×ΓJ(d(j,j′))V×(ui,x(τ+βm),ui′,x′(τ+βm′))×1(lx,i(τ+βm)=j≠j′=lx′,i′(τ+βm′))∑j∈ΓU(2)(ux,i(τ+βm),ux′,i′(τ+βm′))].Next, the functionalB(ΩΛ*) takes into account the bosonic character of the model:
(70)B(ΩΛ*)=∏i∈Λ∏x∈x*(i)zkx,ikx,i.
The factor kx,i-1 in (70) reflects the fact that the starting point of a loop Ωx,i* may be selected among points u(βm,Ωx,i*) arbitrarily.Next, we define the functionalhΛ(ΩΛ*∣x¯Γ¯∖Λ*): for x¯Γ¯∖Λ*={x¯*(j),j∈Γ¯∖Λ}, again assuming that ♯x¯*(j)≤κ. Set
(71)hΛ(ΩΛ*∣x¯Γ¯∖Λ*)=hΛ(ΩΛ*)+hΛ(ΩΛ*∣∣x¯Γ¯∖Λ*).
Here hΛ(ΩΛ*) is as in (67) and
(72)hΛ(ΩΛ*∣∣x¯Γ¯∖Λ*)=∑(i,i′)∈Λ×(Γ¯∖Λ)∑x∈x*(i),x-′∈x¯*(i′)h(x,i),(x-′,i′)(Ωx,i*,(x-′,i′)),
where, in turn,
(73)h(x,i),(x-′,i′)(Ωx,i*,(x-′,i′))=∑0≤m<kx,i‍∫0βdτ×[U(2)(ux,i(τ+βm),x′)×1(lx,i(τ+βm)=i′)+∑j∈ΓJ(d(j,i′))V(ui,x(τ+βm),x′)×1(lx,i(τ+βm)=j)U(2)(ux,i(τ+βm),x′)].
As before, the functionals hΛ(ΩΛ*) and hΛ(ΩΛ*∣∣x¯Γ¯∖Λ*) have a natural interpretation as energies of loop configurations.Finally, as before, the functionalαΛ(ΩΛ*) is the indicator that the collection of loops ΩΛ*={Ωx,i*} does not quit Λ:
(74)αΛ(ΩΛ*)={1,ifΩx,i*(τ)∈M×Λ,∀i∈Λ,x∈x*(i),0≤τ≤βkx,i,0,otherwise.Like above, we invoke known results [11, 15–17] to establish the following statement (again a direct argument can be found in [18]).Lemma 17.
For all finiteΛ⊂Γ and z,β>0 satisfying (22), the partition functions Ξ(Λ) in (27) and Ξ(Λ∣x¯Γ¯∖Λ*) in (27) and (28) admit the representations as converging integrals:
(75)Ξ(Λ)=∫WΛ*dΩΛ*B(ΩΛ*)αΛ(ΩΛ*)exp[-hΛ(ΩΛ*)],(76)Ξ(Λ∣x¯Γ¯∖Λ*)=∫WΛ*dΩΛ*B(ΩΛ*)αΛ(ΩΛ*)exp[-hΛ(ΩΛ*∣∣x¯Γ¯∖Λ*)]
with the ingredients introduced in (60)–(74).Again, we emphasis that the non-zero contribution to the integral in (75) can only come from loop configurations ΩΛ*={Ωx,i*,x∈x*(i),i∈Λ} such that ∀ vertex j∈Λ and τ∈[0,β], the total number of pairs (ux,i(τ+mβ),lx,i(τ+mβ)) with 0≤m<kx,i, and l(τ+mβ)=j does not exceed κ.Remark 18.
The integrals in (75) and (76) represent examples of partition functions which will be encountered in the forthcoming sections. See (96), (97), (101), (103), (105), (107), (111), and (113) below. A general form of such a partition function treated as an integral over a set of loop configurations rather than a trace in a Hilbert space is given in (96) and (97).


## 2.3. The Representation for the RDM Kernels

LetΛ0,Λ be finite sets, Λ0⊂Λ⊂Γ. The construction developed in Section 2.2 also allows us to write a convenient representation for the integral kernels of the RDMs RΛΛ0 (see (31)) and RΛ∣x¯Γ¯∖Λ*Λ0. In accordance with Lemma 11 and the definition of RΛΛ0 in (31), the operator RΛΛ0 acts as an integral operator in ℋ(Λ0):
(77)(RΛΛ0ϕ)(x*0)=∫M*Λ0∏j∈Λ0∏y∈y*(j)v(dy)FΛΛ0(x*0,y*0)ϕ(y*0),
where
(78)FΛΛ0(x*0,y*0):=∫M*Λ∖Λ0∏j∈Λ∖Λ0∏z∈z*(j)v(dz)×FΛ(x*0∨zΛ∖Λ0*,y*0∨zΛ∖Λ0*):=Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0)Ξ(Λ).We employ here and below the notationx*0 and y*0 for particle configurations xΛ0*={x*(i),i∈Λ0} and yΛ0*={y*(j),j∈Λ0} over Λ0. Next, x*0∨zΛ∖Λ0*; y*0∨zΛ∖Λ0* denotes the concatenated configurations over Λ.Similarly, the RDMRΛ∣x¯Γ¯∖Λ*Λ0 is determined by its integral kernel FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0), again admitting the representation
(79)FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0):=Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0∣x¯Γ¯∖Λ*)Ξ(Λ).As in [1], we call FΛΛ0 and FΛ∣x¯Γ¯∖Λ*Λ0 the RDM kernels (in short, RDMKs). The focus of our interest is the numerators Ξ^ΛΛ0(x*0,y*0,Λ∖Λ0) and Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0∣x¯Γ¯∖Λ*) in (78) and (79). To introduce the appropriate representation for these quantities, we need some additional definitions.Definition 19.
Repeating (57)-(58), symbol W¯x*0,y*0* denotes the disjoint union ⋃γ0‍W¯x*0,y*0,γ0* over matchings γ0 between x*0 and y*0. Accordingly, element Ω-*0=Ω-x*0,y*0,γ0*∈W¯x*0,y*0,γ0* yields a collection of paths Ω¯(x,i),γ0(x,i)*∈W¯(x,i),γ0(x,i)* lying in M×Γ. Each path Ω¯(x,i),γ0(x,i)* has time lengths βk(x,i),(y,j), begins at (x,i), and ends up at (y,j)=γ0(x,i), where x∈x*(i), y∈y*(j). Like above, we will use for Ω-*0 the term a path configuration over Λ0. Repeating (63)-(64), we obtain the measures ℙx*0,y*0,γ0* on W¯x*0,y*0,γ0* and ℙx*0,y*0* on W¯x*0,y*0*.The assertion of Lemma20 below again follows directly from known results, in conjunction with calculations of the partial trace trℋΛ∖Λ0 in ℋΛ. The meaning of new ingredients in (79)–(82) is explained below.Lemma 20.
The quantityΞ^ΛΛ0(x*0,y*0;Λ∖Λ0) emerging in (78) is set to be 0 when ♯x*0≠♯y*0. On the other hand, for ♯x*0=♯y*0,
(80)Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0)=∫W¯x*0,y*0*ℙ¯x*0,y*0*(dΩ-*0)B¯(Ω-*0)×αΛ(Ω-*0)1(Ω-*0∈ℱΛ0)×exp[-hΛ0(Ω-*0)]Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0),
where
(81)Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0)=∫WΛ∖Λ0*dΩΛ∖Λ0*B(ΩΛ∖Λ0*)×αΛ(ΩΛ∖Λ0*)1(ΩΛ∖Λ0*∈ℱΛ0)×exp[-hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0)].
Similarly, the quantity Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0∣x¯Γ¯∖Λ*) from (79) vanishes when ♯x*0≠♯y*0. For ♯x*0=♯y*0,
(82)Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0∣x¯Γ¯∖Λ*)=∫W¯x*0,y*0*ℙ¯x*0,y*0*(dΩ-*0)B¯(Ω-*0)×αΛ(Ω-*0)1(Ω-*0∈ℱΛ0)×exp[-hΛ0(Ω-*0∣x¯Γ¯∖Λ*)]×Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0∨x¯Γ¯∖Λ*),
where
(83)Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0∨x¯Γ¯∖Λ*)=∫WΛ∖Λ0*dΩΛ∖Λ0*B(ΩΛ∖Λ0*)αΛ(ΩΛ∖Λ0*)×1(ΩΛ∖Λ0*∈ℱΛ0)×exp[-hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0∨x¯Γ¯∖Λ*)].
These representations hold ∀z,β>0 and finite Λ0⊂Λ⊂Γ.Let us define the functionalsB¯(Ω-*0), αΛ(Ω-*0), 1(Ω-*0∈ℱΛ0), 1(ΩΛ∖Λ0*∈ℱΛ0)hΛ0(Ω-*0), hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0), hΛ0(Ω-*0x¯Γ¯∖Λ*), and hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0∨x¯Γ¯∖Λ*) in (79)–(83). (The functionals B(ΩΛ∖Λ0*) and αΛ(ΩΛ∖Λ0*) are defined as (70) and (74), respectively, replacing Λ with Λ∖Λ0.)To this end, letΩ-*0=Ω-x*0,y*0,γ0*∈W¯x*0,y*0,γ0* be a path configuration represented by a collection of paths Ω¯(x,i),γ0(x,i)*∈W¯(x,i),γ*0(x,i)* (Ω¯x,i* in short), with end points (x,i) and (y,j)=γ0(x,i), of time-length βk(x,i),(y,j). The functional B¯(Ω-*0) is given by
(84)B¯(Ω-*0)=∏i∈Λ‍∏x∈x*(i)‍zk(x,i),(y,j).
The functional αΛ(Ω-*0) is again an indicator:
(85)αΛ(Ω-*0)={1,ifΩ(x,i),γ0(x,i)*(τ)∈M×Λ,∀i∈Λ,x∈x*(i),0≤τ≤βk(x,i),(y,j),0,otherwise.Now let us define the indicator function1(·∈ℱΛ0) in (80)–(83). The factor 1(Ω-*0∈ℱΛ0) equals one if and only if every path Ω¯(x,i),(y,j)* from Ω-*0, of time-length βk(x,i),(y,j), starting at (x,i)∈M×Λ0 and ending up at (y,j)=γ0(x,i)∈M×Λ0 remains in M×(Λ∖Λ0) at the intermediate times βl for l=1,…,k(x,i),(y,j)-1:
(86)Ω¯(x,i),(y,j)*(lβ)∉M×Λ0,∀l=1,…,k(x,i),(y,j)-1,
(when  k(x,i),(y,j)=1, this is not a restriction).Furthermore, suppose thatΩΛ∖Λ0*=ΩxΛ∖Λ0** is a loop configuration over Λ∖Λ0, with the initial/end configuration xΛ∖Λ0*={x*(i),i∈Λ∖Λ0}, represented by a collection of loops Ωx,i*,i∈Λ∖Λ0,x∈x*(i). Then 1(ΩΛ∖Λ0*∈ℱΛ0)=1 if and only if each loop Ωx,i* of time-length βkx,i, beginning and finishing at (x,i)∈M×(Λ∖Λ0), does not enter the set M×Λ0 at times βl for l=1,…,kx,i-1:
(87)Ωx,i*(lβ)∉M×Λ0,∀l=1,…,kx,i-1
(again, if kx,i=1, this is not a restriction).The functionalhΛ0(Ω-*0) in (80) gives the energy of the path configuration Ω-*0 and is introduced similarly to (67), mutatis mutandis. Next, the functional hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0) in (81) represents the energy of the loop configuration ΩΛ∖Λ0* in the potential field generated by the path configuration Ω-*0:
(88)hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0)=hΛ∖Λ0(ΩΛ∖Λ0*)+h(Ω-*0∣∣ΩΛ∖Λ0*).
Here, the summand hΛ∖Λ0(ΩΛ∖Λ0*) yields the energy of the loop configuration ΩΛ∖Λ0*; again confer (67). Further, the term h(Ω-*0∣∣ΩΛ∖Λ0*) yields the energy of interaction between Ω-*0 and ΩΛ∖Λ0*: for a path/loop configurations Ω-*0={Ω¯x,i*}∈W¯x*0,y*0,γ0* and a ΩΛ∖Λ0*={Ωx′,i′*}∈WΛ∖Λ0* we set
(89)h(Ω-*0∣∣ΩΛ∖Λ0*)=∑(i,i′)∈Λ0×(Λ∖Λ0)∑x∈x*(i),x′∈x*(i′)h(x,i),(x′,i′)(Ω¯x,i*,Ωx′,i′*).
Here, for a path Ω¯x,i*=Ω¯(x,i),γ0(x,i)*, of time-length βk(x,i),γ0(x,i), and a loop Ωx′,i′*, of time-length βkx,i,
(90)h(x,i),(x′,i′)(Ω¯x,i*,Ωx′,i′*)=∑0≤m<k(x,i),γ0(x,i)∑0≤m′<kx′,i′∫0βdτ×[∑j∈ΓU(2)(u(τ+βm;Ω¯(x,i),γ0(x,i)*),u(τ+βm′;Ωx′,i′*))×1(l(x,i),γ0(x,i)(τ+βm)=j=lx′,i′(τ+βm′))+∑j,j′∈Γ×Γ‍J(d(j,j′))V(ui,x(τ+βm),ui′,x′(τ+βm′))×1(l(x,i),γ0(x,i)(τ+βm)=j≠j′=lx′,i′(τ+βm′))∑j∈ΓU(2)(u(τ+βm;Ω¯(x,i),γ0(x,i)*),].
Here, in turn, we employ the shortened notation for the positions and indices of the sections Ω¯(x,i),γ0(x,i)*(τ+βm) and Ωx′,i′*(τ+βm′) of Ω¯(x,i),γ0(x,i)* and Ωx′,i′* at times τ+βm and τ+βm′, respectively:
(91)ui,x(τ+βm)=u(τ+βm;Ω¯(x,i),γ0(x,i)*),li,x(τ+βm)=l(τ+βm;Ω¯(x,i),γ0(x,i)*),ui′,x′(τ+βm′)=u(τ+βm′;Ωx′,i′*),li′,x′(τ+βm′)=l(τ+βm′;Ωx′,i′*).Further, the functionalhΛ0(Ω-*0∣x¯Γ¯∖Λ*) in (82) is determined as in (71)–(73), with Ω¯(x,i),γ0(x,i)* instead of Ωx,i*. Next, for hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0∨x¯Γ¯∖Λ*) in (83), we set
(92)hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0∨x¯Γ¯∖Λ*)=hΛ∖Λ0(ΩΛ∖Λ0*)+h(ΩΛ∖Λ0*∣∣Ω-*0∨x¯Γ¯∖Λ*).
Here again, the summand hΛ∖Λ0(ΩΛ∖Λ0*) is determined as in (67). Next, the term hΛ∖Λ0(ΩΛ∖Λ0*∣∣Ω-*0∨x¯Γ¯∖Λ*) is defined similarly to (72)-(73):
(93)hΛ∖Λ0(ΩΛ∖Λ0*∣∣Ω-*0∨x¯Γ¯∖Λ*)=∑(i,i′)∈(Λ∖Λ0)×Λ0∑x∈x*(i),x′∈x*(i′)h(x,i),(x′,i′)(Ωx,i*,Ω¯x′,i′*)+∑i∈Λ∖Λ0∑x∈x*(i),x-′∈x¯*(i′)h(x,i),(x-′,i′)(Ωx,i*,(x-′,i′))
with
(94)h(x,i),(x′,i′)(Ωx,i*,Ω¯x′,i′*)=∑0≤m<kx,i∑0≤m′<k(x′,i′),γ0(x′,i′)×∫0βdτ×[∑j∈Γ‍U(2)(ux,i(τ+βm),ux′,i′(τ+βm′))×1(lx,i(τ+βm)=j=lx′,i′(τ+βm′))+∑(j,j′)∈Γ×ΓJ(d(j,j′))V×(ui,x(τ+βm),ux′,i′(τ+βm′))×1(lx,i(τ+βm)=j≠j′=lx′,i′(τ+βm′))∑j∈Γ‍],h(x,i),(x-′,i′)(Ωx,i*,(x-′,i′))=∑0≤m<kx,i∫0βdτ[(ui,x(τ+βm),x-′)1(lx,i(τ+βm)=j)U(2)(ux,i(τ+βm),x-′)×1(lx,i(τ+βm)=i′)+∑j∈Γ‍J(d(j,i′))V(ui,x(τ+βm),x-′)×1(lx,i(τ+βm)=j)(ui,x(τ+βm),x-′)1(lx,i(τ+βm)=j)U(2)].As before, the functionalshΛ(ΩΛ*) and hΛ(ΩΛ*∣∣x¯Γ¯∖Λ*) have a natural interpretation as energies of loop configurations.Repeating the above observation, non-zero contributions to the integral in (80) come only from pairs (Ω-*0,ΩΛ∖Λ0*) such that ∀j∈Γ and τ∈[0,β], the total number of pairs (u(τ+βm;Ω¯(x,i),γ0(x,i)*),l(τ+βm;Ω¯(x,i),γ0(x,i)*)) with 0≤m<k(x,i),γ0(x,i), i∈Λ0 and x∈x*(i) incident to the paths of the configuration Ω-*0 and pairs (u(τ+βm′;Ωx′,i′*),l(τ+βm′;Ωx′,i′*)) with 0≤m′<k(x′,i′), i′∈Λ∖Λ0 and x′∈x*(i′) incident to the loops of the configuration Ω-Λ∖Λ0* does not exceed κ. Similarly, non-zero contributions to the integral in (82) come only from pairs (Ω-*0,ΩΛ∖Λ0*) such that the above inequality holds when we additionally count points x¯∈x¯*(j).The integralΞ^ΛΛ0(Λ∖Λ0∣Ω-*0) defined in (81) can be considered as a particular (although important) example of a partition function in the volume Λ∖Λ0 with a boundary condition Ω-*0. Note the presence of the subscript Λ indicating that the loops contributing to Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0) can jump within volume Λ only (owing to the indicator functional αΛ). On the other hand, the presence of the indicator functional 1(ΩΛ∖Λ0*∈ℱΛ0) in the integral (reflected in the upperscript Λ0 and the roof sign in the notation Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0)) indicates a particular restriction on the jumps of the loops, forbidding them to visit set Λ0 at intermediate times βm. This is true also for the integral Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0∨x¯Γ¯∖Λ*) in (83): it is a particular example of a partition function in the volume Λ∖Λ0 with a boundary condition Ω-*0∨x¯Γ¯∖Λ*.Other useful types of partition functions areΞΓ0(Λ~∣Ω-*0∨ΩΓ1*) andΞΓ0(Λ~∣Ω-*0∨ΩΓ1*∨x¯Γ2*) where the sets of vertices Λ~, Λ0, Γ0, Γ1, and Γ2 satisfy
(95)Λ~⊂Γ0⊆Γ,Γ1,Γ2⊂Γ∖Λ~,Γ1∩Γ2=∅,Λ0⊂Γ∖(Λ~∪Γ1∪Γ2)
and ♯Λ~,♯Λ0<+∞. Accordingly, Ω-*0 is a (finite) configuration over Λ0, ΩΓ1* a (possibly infinite) loop configuration over Γ1, and x¯Γ2* a (possibly infinite) particle configuration over Γ2. The partition functions ΞΓ0(Λ~∣Ω-*0∨ΩΓ1*) and ΞΓ0(Λ~∣Ω-*0∨ΩΓ1*∨x¯Γ2*) are given by
(96)ΞΓ0(Λ~∣Ω-*0∨ΩΓ1*)=∫WΛ~*dΩΛ~*αΓ0(ΩΛ~*)B(ΩΛ~*)×exp[-hΛ~(ΩΛ~*∣Ω-*0∨ΩΓ1*)],(97)ΞΓ0(Λ~∣Ω-*0∨ΩΓ1*∨x¯Γ2*)=∫WΛ~*dΩΛ~*αΓ0(ΩΛ~*)B(ΩΛ~*)×exp[-hΛ~(ΩΛ~*∣Ω-*0∨ΩΓ1*∨x¯Γ2*)]
with the indicator αΓ0 as in (74). These partition functions, feature loop configurations Ωλ~* formed by loops Ωx,i*, i∈Λ~, which start and finish in Λ~, are confined to Γ0 and move in a potential field generated by Ω-*0∨ΩΓ1*, where Ω-*0={Ω¯(x,i),γ0(x,i)*} and ΩΓ1={Ωx,i*,x∈x*(i),i∈Γ1} or ΩΓ1*∨x¯Γ2*, where x¯Γ2*={x¯*(i),i∈Γ2}. (The latter can be understood as the concatenation of the loop configuration ΩΓ1* over Γ1 and the loop configuration over Γ2 formed by the constant trajectories sitting at points x¯∈x¯*(i), i∈Γ2.) In (96) we assume that, ∀τ∈[0,β] and j∈Γ, the number
(98)♯{(x,i,m):i∈Λ~,l(τ+mβ;Ωx,i*)=j,0≤m<kx,i}+♯{(x,i):i∈Λ0,l(τ+mβ;Ω¯(x,i),γ0(x,i)*)=j,0≤m<k(x,i),γ0(x,i)}+♯{(x,i,m):i∈Γ1,l(τ+mβ;Ωx,i*)=j,0≤m<kx,i}
does not exceed κ. Analogously, in (97) it is assumed that the same is true for the above number plus the cardinality ♯x¯*(j).Such “modified” partition functions will be used in forthcoming sections.


## 2.4.  The FK-DLR MeasureμΛ in a Finite Volume

The Gibbs statesφΛ and φΛ∣x¯Γ¯∖Λ* give rise to probability measures μΛ and μΛ∣x¯Γ¯∖Λ* on the sigma algebra WΛ of subsets of WΛ*. The sigma algebra WΛ is constructed by following the structure of the space WΛ* (a disjoint union of Cartesian products); confer Definition 16. The measures μΛ and μΛ∣x¯Γ¯∖Λ* are determined by their Radon-Nykodym derivative pΛ and pΛ∣x¯Γ¯∖Λ* relative to the measure dΩΛ*:
(99)pΛ(ΩΛ*)∶=μΛ(dΩΛ*)dΩΛ*=1Ξ(Λ)αΛ(ΩΛ*)B(ΩΛ*)×exp[-hΛ(ΩΛ*)],ΩΛ*∈WΛ*,pΛ∣x¯Γ¯∖Λ*(ΩΛ*)∶=μΛ∣x¯Γ¯∖Λ*(dΩΛ*)dΩΛ*1Ξ(Λ∣x¯Γ¯∖Λ*)=αΛ(ΩΛ*)B(ΩΛ*)×exp[-hΛ(ΩΛ*∣x¯Γ¯∖Λ*)],ΩΛ*∈WΛ*.
Given Λ0⊂Λ, the sigma algebra WΛ0 is naturally identified with a sigma subalgebra of WΛ. The restrictions of μΛ to WΛ0 and μΛ∣x¯Γ¯∖Λ* are denoted by μΛΛ0 and μΛ∣x¯Γ¯∖Λ*Λ0; these measures are determined by their Radon-Nikodym derivatives pΛΛ0(ΩΛ0*):=μΛΛ0(dΩΛ0*)/dΩΛ0* and pΛ∣x¯Γ¯∖Λ*Λ0(ΩΛ0*):=μΛ∣x¯Γ¯∖Λ*Λ0(dΩΛ0*)/dΩΛ0*.The first key property of the measuresμΛ and μΛ∣x¯Γ¯∖Λ* is expressed in the so-called FK-DLR equation. We state it as Lemma 21 below; its proof repeats a standard argument used in the classical case for establishing the DLR equation in a finite volume Λ⊂Γ.Lemma 21.
For allz,β>0 satisfying (22), and Λ0⊂Λ′⊂Λ, the probability density pΛΛ0 admits the form:
(100)pΛΛ0(ΩΛ0*)=∫WΛ∖Λ′*qΛ∖Λ′Λ0(ΩΛ0*∣ΩΛ∖Λ′*)μΛΛ∖Λ′(dΩΛ∖Λ′*),
where
(101)qΛ∖Λ′Λ0(ΩΛ0*∣ΩΛ∖Λ′*)=exp[-hΛ0(ΩΛ0*∣ΩΛ∖Λ′*)]×ΞΛ(Λ′∖Λ0∣ΩΛ0*∨ΩΛ∖Λ′*)ΞΛ(Λ′∣ΩΛ∖Λ′*),
and the conditional partition functions ΞΛ(Λ′∖Λ0∣Ω*∨ΩΛ∖Λ′*) andΞΛ(Λ′∣ΩΛ∖Λ′*) are determined as in (96).
Similarly, forpΛ∣x¯Γ¯∖Λ*Λ0 one has:
(102)pΛ∣x¯Γ¯∖Λ*Λ0(ΩΛ0*)=∫WΛ∖Λ′*qΛ∖Λ′Λ0(ΩΛ0*∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*)×μΛ∣x¯Γ¯∖Λ*Λ∖Λ′(dΩΛ∖Λ′*),
where
(103)qΛ∖Λ′Λ0(ΩΛ0*∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*)=exp[-hΛ0(ΩΛ0*∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*)]×ΞΛ(Λ′∖Λ0∣ΩΛ0*∨ΩΛ∖Λ′*∨x¯Γ¯∖Λ*)ΞΛ(Λ′∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*)
and the conditional partition functions ΞΛ(Λ′∖Λ0∣ΩΛ0*∨ΩΛ∖Λ′*∨x¯Γ¯∖Λ*) andΞΛ(Λ′∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*) are determined as in (96).As in [1], (100) and (102) mean that the conditional densities pΛΛ0(ΩΛ0*∣ΩΛ∖Λ′*) and pΛ∣x¯Γ¯∖Λ*Λ0(ΩΛ0*∣ΩΛ∖Λ′*) relative to σ-algebra WΛ∖Λ′ coincide, respectively, with qΛ∖Λ′Λ0(Ω*0∣ΩΛ∖Λ′*), and qΛ∖Λ′Λ0(Ω*0∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*), for μΛΛ∖Λ′—and μΛ∣x¯Γ¯∖Λ*Λ∖Λ′—a.a. ΩΛ∖Λ′*∈WΛ∖Λ′* and a.a. ΩΛ0*∈WΛ0*.As in [1], we call the expressions qΛ∖Λ′Λ0(ΩΛ0*∣ΩΛ∖Λ′*) and qΛ∖Λ′Λ0(ΩΛ0*∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*), as well as the expressions q^Λ∖Λ′Λ0(Ω-*0∣ΩΛ∖Λ′*) and q^Λ∖Λ′Λ0(Ω-*0∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*) appearing below, the (conditional) RDM functionals (in brief, the RDMFs). The same name will be used for the quantity qΛ∖Λ′Λ0(ΩΛ0*∣ΩΛ∖Λ′*) from (110)-(111) and the quantity q^Γ∖ΛΛ0(Ω-*0∣ΩΓ∖Λ*) from (112)-(113).The second property is that the RDMKsFΛΛ0(x*0,y*0) and FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0) are related to the measures μΛ and μΛ∣x¯Γ¯∖Λ′*. Again, the proof of this fact is done by inspection.Lemma 22.
The RDMKFΛΛ0(x*0,y*0) is expressed as follows: ∀Λ0⊂Λ′⊂Λ,
(104)FΛΛ0(x*0,y*0)=∫W¯x0,y0*ℙ¯x0,y0*(dΩ-*0)αΛ(Ω-*0)×B¯(Ω-*0)1(Ω-*0∈ℱΛ0)×∫WΛ∖Λ′*μΛ∖Λ′(dΩΛ∖Λ′*)1(ΩΛ∖Λ′*∈ℱΛ0)×q^Λ∖Λ′Λ0(Ω-*0∣ΩΛ∖Λ′*),
where
(105)q^Λ∖Λ′Λ0(Ω-*0∣ΩΛ∖Λ′*)=exp[-hΛ0(Ω-*0∣ΩΛ∖Λ′*)]×Ξ^ΛΛ0(Λ′∖Λ0∣Ω-*0∨ΩΛ∖Λ′*)ΞΛ(Λ′∣ΩΛ∖Λ′*).
Similarly,(106)FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0)=∫W¯x0,y0*ℙ¯x0,y0*(dΩ-*0)αΛ(Ω-*0)×B¯(Ω-*0)1(Ω-*0∈ℱΛ0)×∫WΛ∖Λ′*μΛ∣x¯Γ¯∖Λ*Λ∖Λ′(dΩΛ∖Λ′*)1(ΩΛ∖Λ′*∈ℱΛ0)×q^Λ∖Λ′Λ0(Ω-*0∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*),
where
(107)q^Λ∖Λ′Λ0(Ω-*0∣ΩΛ∖Λ′*x¯Γ¯∖Λ*)=exp[-hΛ0(Ω-*0∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*)]×Ξ^ΛΛ0(Λ′∖Λ0∣Ω-*0∨ΩΛ∖Λ′*∨x¯Γ¯∖Λ*)ΞΛ(Λ′∣ΩΛ∖Λ′*∨x¯Γ¯∖Λ*).
Here the partition functions Ξ^ΛΛ0(Λ′∖Λ0∣Ω-*0∨ΩΛ∖Λ′*) and Ξ^ΛΛ0(Λ′∖Λ0∣Ω-*0∨ΩΛ∖Λ′*∨x¯Γ¯∖Λ*) are determined in (81) and (83).Remark 23.
Summarizing the above observations, the measuresμΛ and μΛ∣x¯Γ-∖Λ* are concentrated on the subset in WΛ* formed by loop configurations ΩΛ* such that ∀τ∈[0,β], the section ΩΛ*(τ) has ≤κ particles at each vertex i∈Λ.


## 3. The Class of Gibbs States𝔊 for the Fock Space Model

### 3.1. Definition of Class𝔊

In this section we apply the idea from [1] to define the class of states G=Gz,β for the model introduced in Section 2 and state a number of results. These results will hold under condition (34) which is assumed from now on. As in [1], the definition of a state φ∈G is based on the notion of an FK-DLR probability measure μ on the space WΓ*; the class of these measures will be also denoted by G.Definition 24.
SpaceWΓ* is the (infinite) Cartesian product ×i∈ΓW{i}* (cf. (60)); its elements are loop configurations ΩΓ*={Ω*(i),i∈Γ} over Γ. A component Ω*(i) is a finite loop configuration (possibly, empty), with an initial/final particle configuration x*(i)⊂M. Formally, Ω*(i) is a finite collection of loops Ωx,i*, of time-length βkx,i where kx,i=1,2,…, starting and finishing at a point (x,i)∈M×Γ. For reader’s convenience, we repeat (37) for the case under consideration:
(108)Ωx,i*:τ∈[0,βkx,i]⟼(x~(Ωx,i*,τ),i~(Ωx,i*,τ))∈M×Γ,Ωx,i*isca´dla´g;Ωx,i*(0)=Ωx,i*(βkx,i-)=(x,i),Ωx,i*hasfinitelymanyjumpson[0,βkx,i];ifajumpoccursattimeτ,thend[i~(Ωx,i*,τ-),i~(Ωx,i*,τ)]=1.
ByW=WΓ we denote the σ-algebra in WΓ* generated by cylindrical events. Given a subset Γ¯⊂Γ (finite or infinite), we denote by WΓ¯=WΓΓ¯ the σ-subalgebra of W generated by cylindrical events localized in Γ¯. Given a probability measure μ=μΓ on (WΓ*,WΓ), we denote by μΓ¯=μΓΓ¯ the restriction of μ on WΓ¯.Definition 25.
The classG under consideration is formed by measures μ which satisfy the following equation: ∀ finite Λ⊂Γ and Λ0⊆Λ, the probability density:
(109)pΛ0(Ω*0)=pμΛ0(Ω*0):=μΓΛ0(dΩ*0)ν(dΩ*0),Ω*0∈WΛ0*,
is of the form
(110)pΛ0(ΩΛ0*)=∫WΓ∖Λ*qΓ∖ΛΛ0(Ω*0∣ΩΓ∖Λ*)μΓ∖Λ(dΩΓ∖Λ*),
where
(111)qΓ∖ΛΛ0(Ω*0∣ΩΓ∖Λ*)=exp[-hΛ0(Ω*0∣ΩΓ∖Λ*)]×ΞΓ(Λ∖Λ0∣Ω*0∨ΩΓ∖Λ*)ΞΓ(Λ∣ΩΓ∖Λ*),
and the conditional partition functions ΞΓ(Λ∖Λ0∣Ω*0∨ΩΓ∖Λ*) andΞΓ(Λ∣ΩΓ∖Λ*) are determined as in (96).As in [1], (110) means that the conditional density pΛ0∣Γ∖Λ(Ω*0∣ΩΓ∖Λ*), relative to σ-algebra WΓ∖Λ, coincides with qΓ∖ΛΛ0(Ω*0∣ΩΓ∖Λ*), for μΓΓ∖Λ—a.a.ΩΓ∖Λ*∈WΓ∖Λ*β and νΛ0—a.a. Ω*0∈WΛ0*.Remark 26.
The measureμΓ inherits the property from Remark 23 and is concentrated on the subset in WΓ* formed by (infinite) loop configurations ΩΓ* such that, for all τ∈[0,β], the section ΩΛ*(τ) has ≤κ particles at each vertex i∈Λ.Given a measureμ∈G, we associate with it a normalized linear functional φ=φμ on the quasilocal C*-algebra B. First, we set
(112)FΛ0(x*0,y*0)=∫W¯x0,y0*ℙ¯x0,y0*(dΩ-*0)B¯(Ω-*0)1(Ω-*0∈ℱΛ0)×∫WΓ∖Λ*μΓ∖Λ(dΩΓ∖Λ*)1(ΩΓ∖Λ*∈ℱΛ0)×q^Γ∖ΛΛ0(Ω-*0∣ΩΓ∖Λ*),
where
(113)q^Γ∖ΛΛ0(Ω-*0∣ΩΓ∖Λ*)=exp[-hΛ0(Ω-*0∣ΩΓ∖Λ*)]×Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0∨ΩΓ∖Λ*)ΞΓ(Λ∣ΩΓ∖Λ*).This defines a kernelFΛ0(x*0,y*0), x*0,y*0∈M*Λ0, where Λ0⊂Γ is a finite set of sites. It is worth reminding the reader of the presence of the indicator functionals 1(·∈ℱΛ0) in (112) and (113) (in the integral for Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0∨ΩΓ∖Λ*)). These indicators guarantee the compatibility property: ∀ finite Λ0⊂Λ1,
(114)FΛ0(x*0,y*0)=∫M*Λ1∖Λ0∏j∈Λ1∖Λ0∏z∈z*(j)v(dz)×FΛ1(x*0∨zΛ1∖Λ0*,y*0∨zΛ1∖Λ0*).Next, we identify the operatorRΛ0 (a candidate for the RDM in volume Λ0) as an integral operator acting in HΛ0 by
(115)(RΛ0ϕ)(x*0)=∫M*Λ0FΛ0(x*0,y*0)ϕ(y*0)dy*0.Equation (114) implies that
(116)trHΛ1∖Λ0RΛ1=RΛ0.Definition 27.
The functionalφ∈G is identified with the (compatible) family of operators RΛ0. If the operators RΛ0 are positive definite (a property that is not claimed to be automatically fulfilled), we again call it an FK-DLR state in the infinite volume (for given values of activity z and inverse temperature β). To stress the dependence on z and β, we sometimes employ the notation G(z,β).


### 3.2. Theorems on Existence and Properties of FK-DLR States

We are now in position to state results about classG. We assume the conditions on the potentials U(1) and U(2) from the previous section, including the hard-core condition for U(1).Theorem 28.
For allz,β∈(0,+∞) satisfying (22), any limiting Gibbs state φ∈G0 (see Theorem 3) lies in G. Therefore, the class of state G s is nonempty.Theorem 29.
Under condition (22), any FK-DLR state φ∈G is G-invariant, in the sense that, ∀ finite Λ0⊂Γ and ∀g∈G, the RDM RΛ0 satisfies (35). Consequently, (36) holds true.


## 3.1. Definition of Class𝔊

In this section we apply the idea from [1] to define the class of states G=Gz,β for the model introduced in Section 2 and state a number of results. These results will hold under condition (34) which is assumed from now on. As in [1], the definition of a state φ∈G is based on the notion of an FK-DLR probability measure μ on the space WΓ*; the class of these measures will be also denoted by G.Definition 24.
SpaceWΓ* is the (infinite) Cartesian product ×i∈ΓW{i}* (cf. (60)); its elements are loop configurations ΩΓ*={Ω*(i),i∈Γ} over Γ. A component Ω*(i) is a finite loop configuration (possibly, empty), with an initial/final particle configuration x*(i)⊂M. Formally, Ω*(i) is a finite collection of loops Ωx,i*, of time-length βkx,i where kx,i=1,2,…, starting and finishing at a point (x,i)∈M×Γ. For reader’s convenience, we repeat (37) for the case under consideration:
(108)Ωx,i*:τ∈[0,βkx,i]⟼(x~(Ωx,i*,τ),i~(Ωx,i*,τ))∈M×Γ,Ωx,i*isca´dla´g;Ωx,i*(0)=Ωx,i*(βkx,i-)=(x,i),Ωx,i*hasfinitelymanyjumpson[0,βkx,i];ifajumpoccursattimeτ,thend[i~(Ωx,i*,τ-),i~(Ωx,i*,τ)]=1.
ByW=WΓ we denote the σ-algebra in WΓ* generated by cylindrical events. Given a subset Γ¯⊂Γ (finite or infinite), we denote by WΓ¯=WΓΓ¯ the σ-subalgebra of W generated by cylindrical events localized in Γ¯. Given a probability measure μ=μΓ on (WΓ*,WΓ), we denote by μΓ¯=μΓΓ¯ the restriction of μ on WΓ¯.Definition 25.
The classG under consideration is formed by measures μ which satisfy the following equation: ∀ finite Λ⊂Γ and Λ0⊆Λ, the probability density:
(109)pΛ0(Ω*0)=pμΛ0(Ω*0):=μΓΛ0(dΩ*0)ν(dΩ*0),Ω*0∈WΛ0*,
is of the form
(110)pΛ0(ΩΛ0*)=∫WΓ∖Λ*qΓ∖ΛΛ0(Ω*0∣ΩΓ∖Λ*)μΓ∖Λ(dΩΓ∖Λ*),
where
(111)qΓ∖ΛΛ0(Ω*0∣ΩΓ∖Λ*)=exp[-hΛ0(Ω*0∣ΩΓ∖Λ*)]×ΞΓ(Λ∖Λ0∣Ω*0∨ΩΓ∖Λ*)ΞΓ(Λ∣ΩΓ∖Λ*),
and the conditional partition functions ΞΓ(Λ∖Λ0∣Ω*0∨ΩΓ∖Λ*) andΞΓ(Λ∣ΩΓ∖Λ*) are determined as in (96).As in [1], (110) means that the conditional density pΛ0∣Γ∖Λ(Ω*0∣ΩΓ∖Λ*), relative to σ-algebra WΓ∖Λ, coincides with qΓ∖ΛΛ0(Ω*0∣ΩΓ∖Λ*), for μΓΓ∖Λ—a.a.ΩΓ∖Λ*∈WΓ∖Λ*β and νΛ0—a.a. Ω*0∈WΛ0*.Remark 26.
The measureμΓ inherits the property from Remark 23 and is concentrated on the subset in WΓ* formed by (infinite) loop configurations ΩΓ* such that, for all τ∈[0,β], the section ΩΛ*(τ) has ≤κ particles at each vertex i∈Λ.Given a measureμ∈G, we associate with it a normalized linear functional φ=φμ on the quasilocal C*-algebra B. First, we set
(112)FΛ0(x*0,y*0)=∫W¯x0,y0*ℙ¯x0,y0*(dΩ-*0)B¯(Ω-*0)1(Ω-*0∈ℱΛ0)×∫WΓ∖Λ*μΓ∖Λ(dΩΓ∖Λ*)1(ΩΓ∖Λ*∈ℱΛ0)×q^Γ∖ΛΛ0(Ω-*0∣ΩΓ∖Λ*),
where
(113)q^Γ∖ΛΛ0(Ω-*0∣ΩΓ∖Λ*)=exp[-hΛ0(Ω-*0∣ΩΓ∖Λ*)]×Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0∨ΩΓ∖Λ*)ΞΓ(Λ∣ΩΓ∖Λ*).This defines a kernelFΛ0(x*0,y*0), x*0,y*0∈M*Λ0, where Λ0⊂Γ is a finite set of sites. It is worth reminding the reader of the presence of the indicator functionals 1(·∈ℱΛ0) in (112) and (113) (in the integral for Ξ^ΛΛ0(Λ∖Λ0∣Ω-*0∨ΩΓ∖Λ*)). These indicators guarantee the compatibility property: ∀ finite Λ0⊂Λ1,
(114)FΛ0(x*0,y*0)=∫M*Λ1∖Λ0∏j∈Λ1∖Λ0∏z∈z*(j)v(dz)×FΛ1(x*0∨zΛ1∖Λ0*,y*0∨zΛ1∖Λ0*).Next, we identify the operatorRΛ0 (a candidate for the RDM in volume Λ0) as an integral operator acting in HΛ0 by
(115)(RΛ0ϕ)(x*0)=∫M*Λ0FΛ0(x*0,y*0)ϕ(y*0)dy*0.Equation (114) implies that
(116)trHΛ1∖Λ0RΛ1=RΛ0.Definition 27.
The functionalφ∈G is identified with the (compatible) family of operators RΛ0. If the operators RΛ0 are positive definite (a property that is not claimed to be automatically fulfilled), we again call it an FK-DLR state in the infinite volume (for given values of activity z and inverse temperature β). To stress the dependence on z and β, we sometimes employ the notation G(z,β).


## 3.2. Theorems on Existence and Properties of FK-DLR States

We are now in position to state results about classG. We assume the conditions on the potentials U(1) and U(2) from the previous section, including the hard-core condition for U(1).Theorem 28.
For allz,β∈(0,+∞) satisfying (22), any limiting Gibbs state φ∈G0 (see Theorem 3) lies in G. Therefore, the class of state G s is nonempty.Theorem 29.
Under condition (22), any FK-DLR state φ∈G is G-invariant, in the sense that, ∀ finite Λ0⊂Γ and ∀g∈G, the RDM RΛ0 satisfies (35). Consequently, (36) holds true.


## 4. Proof of Theorems3, 6, 28, and 29

### 4.1. Proof of Theorems3 and 28

The proof is based on the same approach as that used in [1]. First, given Λ0⊂Γ, we establish compactness of the sequence of the RDMKs FΛΛ0(x*0,y*0) and FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0) (see (78)–(83)) as functions of variables x*0={x*0(i)},y*0={y*0(i)}∈M*Λ0, with
(117)♯xΛ*0=♯yΛ*0,♯x*0(i),♯y*0(i)<κ,i∈Λ,
when Λ↗Γ. Then we use Lemma 1.1 from [1] to derive that the sequence of the RDMs RΛΛ0 and RΛ∣x¯Γ¯∖Λ*Λ0 is compact in the trace-norm operator topology in ℋΛ0.To verify compactness of the RDMKsFΛΛ0(x*0,y*0) and FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0) we, again as in [1], use the Ascoli-Arzela theorem, which requires the properties of uniform boundedness and equicontinuity. These properties follow from the following.Lemma 30.
(i) Under condition (22) the RDMKs FΛΛ0(x*0,y*0) and FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0) admit the bounds
(118)FΛΛ0(x*0,y*0),FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0)≤[(κ♯Λ0)!](p^M)κ♯Λ0Φ♯Λ0,
where
(119)Φ=∑k≥1zkexp(kΘ),withΘ=κβ(U¯(1)+κU¯(2)+κJ¯(1)V¯).
(Note that Φ<∞ under the assumption (22)) Let pM(k) yields the supremum of the transition function over time kβ for Brownian motion on the torus M:
(120)pM(k)=supx,y∈Mpkβ(x,y)=pkβ(0,0),
and p^M=supk≥1pM(k). Finally, the upper-bound values U¯(1), U¯(2), J¯(1), and V¯ have been determined in (14), (15), and (18).
(ii) The gradients of the RDMKsFΛΛ0(x*0,y*0) and FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0) satisfy ∀i∈Λ and x∈x*0(i), y∈y*0(i),
(121)|∇xFΛΛ0(x*0,y*0)|,|∇xFΛ∣x¯Γ¯∖Λ*Λ0(x0,y0)||∇yFΛΛ0(x*0,y*0)|,|∇yFΛ∣x¯Γ¯∖Λ*Λ0(x0,y0)|≤(♯Λ0)ΘΦ′p^M[(κ♯Λ0)!](p^M)κ♯Λ0Φ♯Λ0,
where
(122)Φ′=∑k≥1kzkexp(kΘ).
(Again, Φ′<∞ under the condition (22).) The ingredients p^M and U¯(1), U¯(2), J¯(1), and V¯ as in statement (i).Proof of Lemma30.
First, observe thatp^M<∞ on a compact manifold. Bound (119) is established in a direct fashion. First, we majorize the energy
(123)hΛ0(Ω-*0∣ΩΛ∖Λ0*)=hΛ0(Ω-*0)+h(ΩΛ∖Λ0*∣∣Ω-*0)
contributing to the RHS in (80) and (81) and the energy
(124)hΛ0(Ω-*0∣ΩΛ∖Λ0*∨x¯Γ¯∖Λ*)=hΛ0(Ω-*0∣x¯Γ¯∖Λ*)+h(ΩΛ∖Λ0*∣∣Ω-*0)
contributing to the RHS in (82) and (83). This yields the factor
(125)∏ω¯∈Ω-*0zk(ω¯*)exp[k(ω¯*)Θ].
Next, we majorize the integral =∫W-x*0,y*0*ℙ¯x*0,y*0*(dΩ-*0) in (80) and (82); this gives the factor
(126)(♯Λ0)p^M[(κ♯Λ0)!](p^M)κ♯Λ0Φ♯Λ0.
The aftermath are the ratios (78) and (79) with x*0=y*0=∅; they do not exceed 1.
Passing to (121), let us discuss the gradients ∇x only. (The gradients in the entries of ∇y are included by symmetry.) The gradient in (121), of course, affects only the numerators Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0) and Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0∣x¯Γ¯∖Λ*) in (78) and (79). The bounds (121) are done essentially as in [1]. For definiteness, we discuss the case of the RDMK FΛΛ0(x*0,y*0); the RDMK FΛ∣x¯Γ¯∖Λ*Λ0(x0,y0) is treated similarly. There are two contributions into the gradient: one comes from varying the measure ℙ¯x*0,y*0*(dΩ-*0), and the other from varying the functional exp[-hΛ0(Ω-*0)-hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0)].
The first contribution can again be uniformly bounded in terms of the constantp^M. The detailed argument, as in [1], includes a deformation of a trajectory and is done similarly to [1] (the presence of jumps does not change the argument because p^M yields a uniform bound in (43)).
The second contribution yields, again as in [1], an expression of the form:
(127)∫W-x*0,y*0*ℙ¯x*0,y*0*(dΩ-*0)×∑i∈Λ0∑x∈x*0(i)h~x,i(Ω¯(x,i),γ0(x,i)*,ΩΛ∖Λ0*)×exp[-hΛ0(Ω-*0)-hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0)],
where the functional h~x,i(Ω¯(x,i),γ0(x,i)*,ΩΛ∖Λ0*) is uniformly bounded. Combining this an upper bound similar to (119) yields the desired estimate for the gradients in (121).Hence, we can guarantee that the RDMsRΛΛ0 and RΛ∣x¯Γ¯∖Λ*Λ0 converge to a limiting RDM RΛ0 along a subsequence in Λ↗Γ. The diagonal process yields convergence for every finite Λ0⊂Γ. A parallel argument leads to compactness of the measures μΛΛ0 for any given Λ0 as Λ↗Γ. We only give here a sketch of the corresponding argument, stressing differences with its counterpart in [1].In the probabilistic terminology, measuresμΛ represent random marked point fields on M×Γ with marks from the space W*=W0* where W0*=⋃k≥1W0kβ and W0kβ is the space of loops of time-length kβ starting and finishing at 0∈M and exhibiting jumps, that is, changes of the index. (The space Wx,i* introduced in Definitions 7 and 13 can be considered as a copy of W* placed at site i∈Γ and point x∈M.) The measure μΛΛ0 describes the restriction of μΛ to volume Λ0 (i.e., to the sigma algebra 𝔚Λ0*) and is given by its Radon-Nikodym derivative pΛΛ0 relative to the reference measure dΩΛ0* on WΛ0* (cf. (66), (100)). The reference measure is sigma-finite. Moreover, under the condition (22), the value pΛΛ0(ΩΛ0*) is uniformly bounded (in both Λ↗Γ and ΩΛ0*∈WΛ0*). This enables us to verify tightness of the family of measures {μΛΛ0,Λ↗Γ} and apply the Prokhorov theorem. Next, we use the compatibility property of the limit-point measures μΓΛ0 and apply the Kolmogorov theorem. This establishes the existence of the limit-point measure μΓ.By construction, and owing to Lemmas21 and 22, the limiting family {RΛ0} yields a state belonging to the class G. This completes the proof of Theorems 3 and 28.


### 4.2. Proof of Theorems6 and 29

The assertion of Theorem6 is included in Theorem 29. Therefore, we will focus on the proof of the latter. The proof based on the analysis of the conditional RDMFs qΓ∖ΛΛ0(Ω-*0∣ΩΓ∖Λ*) and q^Γ∖ΛΛ0(Ω-*0∣ΩΓ∖Λ*) introduced in (111) and (112). For definiteness, we assume that vertex o∈Λ0, so that Λ0 lies in the ball Λn for n large enough. As in [1], the problem is reduced to checking that ∀z,β∈(0,∞) satisfying (22), g∈G and finite Λ0⊂Γ,
(128)limn→∞q^Γ∖Λ(n)Λ0(gΩ-*0∣ΩΓ∖Λ(n)*)q^Γ∖Λ(n)Λ0(Ω-*0∣ΩΓ∖Λ(n)*)=1;
here we need to establish this convergence (128) uniformly in the argument ΩΓ∖Λ(n)*={Ω*(i),i∈Γ∖Λ(n)} with ♯Ω*(i)≤κ and in Ω-*0 outside a set of the ℙ¯x0,y0*β measure tending to 0 as n→∞. The latter is formed by path configurations Ω-*0 that contain trajectories visiting sites i∈Γ∖Λ(r¯(n)) where r¯(n) grows with n; see Lemma 31 below. The action of g upon a path configuration Ω-*0={Ω¯(x,i),γ0(x,i)*,i∈Λ0,x∈x*0} is defined by
(129)gΩ-*0={gΩ¯(x,i),γ0(x,i)*},where(gΩ¯(x,i),γ0(x,i)*)(τ)=g(Ω¯(x,i),γ0(x,i)*(τ)).We want to establish that∀a∈(1,∞), for any n large enough, the conditional RDMFs satisfy
(130)aq^Γ∖Λ(n)Λ0(gΩ-*0∣ΩΓ∖Λ(n)*)+aq^Γ∖Λ(n)Λ0(g-1Ω-*0∣ΩΓ∖Λ(n)*)≥2q^Γ∖Λ(n)Λ0(Ω-*0∣ΩΓ∖Λ(n)*).As in [1], we deduce (130) with the help of a special construction of “tuned” actions gΛ(n)∖Λ0 on loop configurations ωΛ(n)∖Λ0 (over which there is integration performed in the numerators
(131)Ξ^ΓΛ0(Λ(n)∖Λ0∣gΩ-*0∨ΩΓ∖Λ(n)*),Ξ^ΓΛ0(Λ(n)∖Λ0∣g-1Ω-*0∨ΩΓ∖Λ(n)*)
in the expression for q^Γ∖Λ(n)Λ0(gΩ-*0∣ΩΓ∖Λ(n)*) and q^Γ∖Λ(n)Λ0(g-1Ω-*0∣ΩΓ∖Λ(n)*)). The tuning in gΛ(n)∖Λ0 is chosen so that it approaches e (or the d-dimensional zero vector in the additive form of writing), the neutral element of G, while we move from Λ0 towards Γ∖Λ(n).Formally, (130) follows from the estimate (132) below: ∀ finite Λ0⊂Γ,Ω-*0∈W¯Λ(n), g∈G and a∈(1,∞), for any n large enough, ∀ΩΛ(n)∖Λ0*={Ω*(i),i∈Λ(n)∖Λ0} and ΩΓ∖Λ(n)*={Ω*(i),i∈Γ∖Λ(n)} with ♯Ω*(i)≤κ,
(132)a2exp[-hΛ(n)((gΩ-*0)∨(gΛ(n)∖Λ0ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)]+a2exp[((g-1Ω-*0)∨(gΛ(n)∖Λ0-1ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)-hΛ(n)×((g-1Ω-*0)∨(gΛ(n)∖Λ0-1ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)]≥exp[-hΛ(n)(Ω-*0∨ΩΛ(n)∖Λ0*∣ΩΓ∖Λ(n)*)].In (132), the loop configuration gΛ(n)∖Λ0ΩΛ(n)∖Λ0* is determined by specifying its temporal section {gΛ(n)∖Λ0Ωx,i*(τ+βm),i∈Λ(n)∖Λ0,x∈x*(i),0≤m<kx,i}. That is, we need to specify the sections:
(133)(u(τ+βm;gΛ(n)∖Λ0Ωx,i*),l(τ+βm;gΛ(n)∖Λ0Ωx,i*))
for loops Ωx,i* constituting gΛ(n)∖Λ0ΩΛ(n)∖Λ0*. To this end we set
(134)u(τ+βm;gΛ(n)∖Λ0Ωx,i*)=gj(n)u(τ+βm;Ωx,i*)ifl(τ+βm;gΛ(n)∖Λ0Ωx,i*)=j.
In other words, we apply the action gj(n) to the temporal sections of all loops Ωx,i* located at vertex j at a given time, regardless of position of their initial points (x,i) in Λ(n)∖Λ0.Observe that (130) is deduced from (132) by integrating in dΩΛ(n)∖Λ0* and normalizing by ΞΛ(n)∖Λ0(ΩΓ∖Λ(n)*); see (80) with Λ′=Λ(n). (The Jacobian of the map ΩΛ(n)∖Λ0*↦gΛ(n)∖Λ0ΩΛ(n)∖Λ0* equals 1.)Thus, our aim becomes proving (132). The tuned family gΛ(n)∖Λ0 is composed of individual actions gj(n)∈G:
(135)gΛ(n)∖Λ0={gj(n),j∈Λ(n)∖Λ0}.
Elements gj(n) are powers (multiples, in the additive parlance) of element g∈G figuring in (128)–(132) (resp., of the corresponding vector θ_∈M; cf. (9)) and defined as follows. Let θ_j(n) denote the vector from M corresponding to gj(n), and we select positive integer values r¯(n)=⌈log(1+n)⌉ and set
(136)θ_j(n)=θ_υ(n,j),
where
(137)υ(n,j)={1,d(o,j)≤r¯(n),ϑ(d(j,o)-r¯(n),n-r¯(n)),d(o,j)>r¯(n).
In turn, the function ϑ is chosen to satisfy
(138)ϑ(a,b)=1(a≤0)+1(0<a<b)Q(b)×∫abz(u)du,a,b∈ℝ,
with
(139)Q(b)=∫0b‍ζ(u)du~loglogb,whereζ(u)=1(u≤2)+1(u>2)1ulnu,b>0.Lemma 31.
Givenz,β∈(0,∞) satisfying (22) and a finite set Λ0, there exists a constant C∈(0,∞) such that ∀x0,y0∈M*Λ0, the set of path configurations Ω-*0∈W¯x*0,y*0* with hΛ0(Ω-*0)<+∞ which include trajectories visiting points in Γ∖Λ(r¯(n)) has the ℙ¯x0,y0* measure that does not exceed C/(⌈(1+r(n))⌉)!.Proof of Lemma31.
The condition thathΛ0(Ω-*0)<+∞ implies that the total number of sub trajectories of length β in Ω-*0 does not exceed κ×♯Λ0 which is a fixed value in the context of the lemma. Each such trajectory has a Poisson number of jumps; this produces the factor 1/(⌈(1+r(n))⌉)!.Back to the proof of Theorem29: let gΛ(n)∖Λ0-1 be the collection of the inverse elements:
(140)gΛ(n)∖Λ0-1={gj(n)-1,j∈Λ(n)∖Λ0}.
The vectors corresponding to gj(n)-1 are -θ_j(n)∈M. We will use this specification for gj(n) and gj(n)-1 for j∈Λ(n), or even for j∈Γ, as it agrees with the requirement that gj(n)≡g when j∈Λ0 and gj(n)≡e for j∈Γ∖Λ(n). Accordingly, we will use the notation gΛ(n)={gj(n),j∈Λ(n)}.Observe that the tuned familygΛ(n)∖Λ0 does not change the contribution into the energy functional hΛ(n)∣Γ∖Λ(n) coming from potentials U(1) and U(2): it affects only contributions from potential V.The Taylor formula for functionV, together with the above identification of vectors θ_j(n), gives
(141)|V(gj(n)x,gj′(n)x′)+V(gj(n)-1x,gj′(n)-1x′)-2V(x,x′)|≤C|θ_|2|υ(n,j)-υ(n,j′)|2V¯,x,x′∈M.
Here C∈(0,∞) is a constant, |θ_| stands for the norm of the vector θ_ representing the element g, and the value V¯ is taken from (14).Next, the square|υ(n,j)-υ(n,j′)|2 can be specified as
(142)|υ(n,j)-υ(n,j′)|2={0,ifd(j,o),d(j′,o)≤r¯(n),0,ifd(j,o),d(j′,o)≥n[(d(j′,o)-r¯(n),n-r¯(n))ϑ(d(j,o)-r¯(n),n-r¯(n))-ϑ(d(j′,o)-r¯(n),n-r¯(n))]2,ifr¯(n)<d(j,o),d(j′,o)≤n,ϑ(d(j,o)-r¯(n),n-r¯(n))2,ifr¯(n)<d(j,o)≤n,d(j′,o)∈]r¯(n),n[,ϑ(d(j′,o)-r¯(n),n-r¯(n))2,ifr¯(n)<d(j′,o)≤n,d(j,o)∈]r¯(n),n[.By using convexity of the function exp and (141), ∀a>1,
(143)a2exp[-hΛ(n)(gΛ(n)(Ω-*0∨ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)]+a2exp[-hΛ(n)(gΛ(n)-1(Ω-*0∨ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)]≥aexp[-12hΛ(n)∣Γ∖Λ(n)×(gΛ(n)(Ω-*0∨ΩΛ(n)∖Λ0*),ΩΓ∖Λ(n)*)-12hΛ(n)(gΛ(n)-1(Ω-*0∨ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)]≥aexp[-hΛ(n)(Ω-*0∨ΩΛ(n)∖Λ0*∣ΩΓ∖Λ(n)*)]e-CΥ/2,
where
(144)Υ=Υ(n,g)=βκ2×∑(j,j′)∈Λ(n)×ΓJ(d(j,j′))|υ(n,j)-υ(n,j′)|2.
The next observation is that
(145)Υ≤3βκ2|θ_|2×∑(j,j′)∈Λ(n)×Γ1(d(j,o)≤d(j′,0))J(d(j,j′))×[(j′,o)ϑ(d(j,o)-r¯(n),n-r¯(n))-ϑ(d(j′,o)-r¯(n),n-r¯(n))]2,
where, owing to the triangle inequality, for all j,j′:d(j,o)≤d(j′,o)(146)0≤ϑ(d(j,o)-r¯(n),n-r¯(n))-ϑ(d(j′,o)-r¯(n),n-r¯(n))≤d(j,j′)ζ(d(j,o)-r¯)Q(n-r¯(n)).
This yields
(147)Υ≤βκ23|θ_|2Q(n-r¯(n))2×∑(j,j′)∈Λ(n)×ΓJ(d(j,j′))d(j,j′)2ζ(d(j,0)-r¯(n))2≤3|θ_|2Q(n-r¯(n))2[supj∈Γ∑j′∈ΓJj,j′d(j,j′)2]×∑j∈Λn+r0ζ(d(j,0)-r¯(n))2,
where function ζ is determined in (139).Therefore, it remains to estimate the sum∑j∈Λn+r0ζ(d(j,0)-r¯(n))2. To this end, observe that uζ(u)<1 when u∈(3,∞). The next remark is that the number of sites in the sphere Σn grows linearly with n. Consequently,
(148)∑j∈Λn+r0ζ(d(j,o)-r¯(n))2=∑1≤k≤n+r0ζ(k-r¯(n))×∑j∈Σkζ(k-r¯(n))≤C0∑1≤k≤n+r0ζ(k-r¯(n))≤C1Q(n+r0-r¯(n)),Υ≤CQ(n-r¯(n))⟶∞,asn⟶∞.Therefore, givena>1 for n large enough, the term ae-CΥ/2 in the RHS of (145) becomes >1. Hence,
(149)a2exp[-hΛ(n)(gΛ(n)(Ω-*0∨ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)]+a2exp[-hΛ(n)(gΛ(n)-1(Ω-*0∨ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)]≥exp[-hΛ(n)(Ω-*0∨ΩΛ(n)∖Λ0*∣ΩΓ∖Λ(n)*)].
Equation (149) implies that the quantity
(150)qΛ0∣Γ∖Λ(n)(Ω-*0∣ΩΓ∖Λ(n))=∫WΛ(n)∖Λ0dΩΛ(n)∖Λ0*×exp[-hΛ0(Ω-*0∨ΩΛ(n)∖Λ0*∣ΩΓ∖Λ0*)]ΞΛ(n)(ΩΓ∖Λ(n)*),
obeys
(151)limn→∞[qβΛ0∣Γ∖Λ(n)(gΩ-*0∣ΩΓ∖Λ(n)*)+qβΛ0∣Γ∖Λ(n)(g-1Ω-*0∣ΩΓ∖Λ(n)*)]≥2limn→∞qβΛ0∣Γ∖Λ(n)(Ω-*0∣ΩΓ∖Λ(n)*)
uniformly in boundary condition ωΓ∖Λ(n). Integrating (151) dμΓΓ∖Λ(n)(ωΓ∖Λ(n)) yields (132).


## 4.1. Proof of Theorems3 and 28

The proof is based on the same approach as that used in [1]. First, given Λ0⊂Γ, we establish compactness of the sequence of the RDMKs FΛΛ0(x*0,y*0) and FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0) (see (78)–(83)) as functions of variables x*0={x*0(i)},y*0={y*0(i)}∈M*Λ0, with
(117)♯xΛ*0=♯yΛ*0,♯x*0(i),♯y*0(i)<κ,i∈Λ,
when Λ↗Γ. Then we use Lemma 1.1 from [1] to derive that the sequence of the RDMs RΛΛ0 and RΛ∣x¯Γ¯∖Λ*Λ0 is compact in the trace-norm operator topology in ℋΛ0.To verify compactness of the RDMKsFΛΛ0(x*0,y*0) and FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0) we, again as in [1], use the Ascoli-Arzela theorem, which requires the properties of uniform boundedness and equicontinuity. These properties follow from the following.Lemma 30.
(i) Under condition (22) the RDMKs FΛΛ0(x*0,y*0) and FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0) admit the bounds
(118)FΛΛ0(x*0,y*0),FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0)≤[(κ♯Λ0)!](p^M)κ♯Λ0Φ♯Λ0,
where
(119)Φ=∑k≥1zkexp(kΘ),withΘ=κβ(U¯(1)+κU¯(2)+κJ¯(1)V¯).
(Note that Φ<∞ under the assumption (22)) Let pM(k) yields the supremum of the transition function over time kβ for Brownian motion on the torus M:
(120)pM(k)=supx,y∈Mpkβ(x,y)=pkβ(0,0),
and p^M=supk≥1pM(k). Finally, the upper-bound values U¯(1), U¯(2), J¯(1), and V¯ have been determined in (14), (15), and (18).
(ii) The gradients of the RDMKsFΛΛ0(x*0,y*0) and FΛ∣x¯Γ¯∖Λ*Λ0(x*0,y*0) satisfy ∀i∈Λ and x∈x*0(i), y∈y*0(i),
(121)|∇xFΛΛ0(x*0,y*0)|,|∇xFΛ∣x¯Γ¯∖Λ*Λ0(x0,y0)||∇yFΛΛ0(x*0,y*0)|,|∇yFΛ∣x¯Γ¯∖Λ*Λ0(x0,y0)|≤(♯Λ0)ΘΦ′p^M[(κ♯Λ0)!](p^M)κ♯Λ0Φ♯Λ0,
where
(122)Φ′=∑k≥1kzkexp(kΘ).
(Again, Φ′<∞ under the condition (22).) The ingredients p^M and U¯(1), U¯(2), J¯(1), and V¯ as in statement (i).Proof of Lemma30.
First, observe thatp^M<∞ on a compact manifold. Bound (119) is established in a direct fashion. First, we majorize the energy
(123)hΛ0(Ω-*0∣ΩΛ∖Λ0*)=hΛ0(Ω-*0)+h(ΩΛ∖Λ0*∣∣Ω-*0)
contributing to the RHS in (80) and (81) and the energy
(124)hΛ0(Ω-*0∣ΩΛ∖Λ0*∨x¯Γ¯∖Λ*)=hΛ0(Ω-*0∣x¯Γ¯∖Λ*)+h(ΩΛ∖Λ0*∣∣Ω-*0)
contributing to the RHS in (82) and (83). This yields the factor
(125)∏ω¯∈Ω-*0zk(ω¯*)exp[k(ω¯*)Θ].
Next, we majorize the integral =∫W-x*0,y*0*ℙ¯x*0,y*0*(dΩ-*0) in (80) and (82); this gives the factor
(126)(♯Λ0)p^M[(κ♯Λ0)!](p^M)κ♯Λ0Φ♯Λ0.
The aftermath are the ratios (78) and (79) with x*0=y*0=∅; they do not exceed 1.
Passing to (121), let us discuss the gradients ∇x only. (The gradients in the entries of ∇y are included by symmetry.) The gradient in (121), of course, affects only the numerators Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0) and Ξ^ΛΛ0(x*0,y*0;Λ∖Λ0∣x¯Γ¯∖Λ*) in (78) and (79). The bounds (121) are done essentially as in [1]. For definiteness, we discuss the case of the RDMK FΛΛ0(x*0,y*0); the RDMK FΛ∣x¯Γ¯∖Λ*Λ0(x0,y0) is treated similarly. There are two contributions into the gradient: one comes from varying the measure ℙ¯x*0,y*0*(dΩ-*0), and the other from varying the functional exp[-hΛ0(Ω-*0)-hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0)].
The first contribution can again be uniformly bounded in terms of the constantp^M. The detailed argument, as in [1], includes a deformation of a trajectory and is done similarly to [1] (the presence of jumps does not change the argument because p^M yields a uniform bound in (43)).
The second contribution yields, again as in [1], an expression of the form:
(127)∫W-x*0,y*0*ℙ¯x*0,y*0*(dΩ-*0)×∑i∈Λ0∑x∈x*0(i)h~x,i(Ω¯(x,i),γ0(x,i)*,ΩΛ∖Λ0*)×exp[-hΛ0(Ω-*0)-hΛ∖Λ0(ΩΛ∖Λ0*∣Ω-*0)],
where the functional h~x,i(Ω¯(x,i),γ0(x,i)*,ΩΛ∖Λ0*) is uniformly bounded. Combining this an upper bound similar to (119) yields the desired estimate for the gradients in (121).Hence, we can guarantee that the RDMsRΛΛ0 and RΛ∣x¯Γ¯∖Λ*Λ0 converge to a limiting RDM RΛ0 along a subsequence in Λ↗Γ. The diagonal process yields convergence for every finite Λ0⊂Γ. A parallel argument leads to compactness of the measures μΛΛ0 for any given Λ0 as Λ↗Γ. We only give here a sketch of the corresponding argument, stressing differences with its counterpart in [1].In the probabilistic terminology, measuresμΛ represent random marked point fields on M×Γ with marks from the space W*=W0* where W0*=⋃k≥1W0kβ and W0kβ is the space of loops of time-length kβ starting and finishing at 0∈M and exhibiting jumps, that is, changes of the index. (The space Wx,i* introduced in Definitions 7 and 13 can be considered as a copy of W* placed at site i∈Γ and point x∈M.) The measure μΛΛ0 describes the restriction of μΛ to volume Λ0 (i.e., to the sigma algebra 𝔚Λ0*) and is given by its Radon-Nikodym derivative pΛΛ0 relative to the reference measure dΩΛ0* on WΛ0* (cf. (66), (100)). The reference measure is sigma-finite. Moreover, under the condition (22), the value pΛΛ0(ΩΛ0*) is uniformly bounded (in both Λ↗Γ and ΩΛ0*∈WΛ0*). This enables us to verify tightness of the family of measures {μΛΛ0,Λ↗Γ} and apply the Prokhorov theorem. Next, we use the compatibility property of the limit-point measures μΓΛ0 and apply the Kolmogorov theorem. This establishes the existence of the limit-point measure μΓ.By construction, and owing to Lemmas21 and 22, the limiting family {RΛ0} yields a state belonging to the class G. This completes the proof of Theorems 3 and 28.


## 4.2. Proof of Theorems6 and 29

The assertion of Theorem6 is included in Theorem 29. Therefore, we will focus on the proof of the latter. The proof based on the analysis of the conditional RDMFs qΓ∖ΛΛ0(Ω-*0∣ΩΓ∖Λ*) and q^Γ∖ΛΛ0(Ω-*0∣ΩΓ∖Λ*) introduced in (111) and (112). For definiteness, we assume that vertex o∈Λ0, so that Λ0 lies in the ball Λn for n large enough. As in [1], the problem is reduced to checking that ∀z,β∈(0,∞) satisfying (22), g∈G and finite Λ0⊂Γ,
(128)limn→∞q^Γ∖Λ(n)Λ0(gΩ-*0∣ΩΓ∖Λ(n)*)q^Γ∖Λ(n)Λ0(Ω-*0∣ΩΓ∖Λ(n)*)=1;
here we need to establish this convergence (128) uniformly in the argument ΩΓ∖Λ(n)*={Ω*(i),i∈Γ∖Λ(n)} with ♯Ω*(i)≤κ and in Ω-*0 outside a set of the ℙ¯x0,y0*β measure tending to 0 as n→∞. The latter is formed by path configurations Ω-*0 that contain trajectories visiting sites i∈Γ∖Λ(r¯(n)) where r¯(n) grows with n; see Lemma 31 below. The action of g upon a path configuration Ω-*0={Ω¯(x,i),γ0(x,i)*,i∈Λ0,x∈x*0} is defined by
(129)gΩ-*0={gΩ¯(x,i),γ0(x,i)*},where(gΩ¯(x,i),γ0(x,i)*)(τ)=g(Ω¯(x,i),γ0(x,i)*(τ)).We want to establish that∀a∈(1,∞), for any n large enough, the conditional RDMFs satisfy
(130)aq^Γ∖Λ(n)Λ0(gΩ-*0∣ΩΓ∖Λ(n)*)+aq^Γ∖Λ(n)Λ0(g-1Ω-*0∣ΩΓ∖Λ(n)*)≥2q^Γ∖Λ(n)Λ0(Ω-*0∣ΩΓ∖Λ(n)*).As in [1], we deduce (130) with the help of a special construction of “tuned” actions gΛ(n)∖Λ0 on loop configurations ωΛ(n)∖Λ0 (over which there is integration performed in the numerators
(131)Ξ^ΓΛ0(Λ(n)∖Λ0∣gΩ-*0∨ΩΓ∖Λ(n)*),Ξ^ΓΛ0(Λ(n)∖Λ0∣g-1Ω-*0∨ΩΓ∖Λ(n)*)
in the expression for q^Γ∖Λ(n)Λ0(gΩ-*0∣ΩΓ∖Λ(n)*) and q^Γ∖Λ(n)Λ0(g-1Ω-*0∣ΩΓ∖Λ(n)*)). The tuning in gΛ(n)∖Λ0 is chosen so that it approaches e (or the d-dimensional zero vector in the additive form of writing), the neutral element of G, while we move from Λ0 towards Γ∖Λ(n).Formally, (130) follows from the estimate (132) below: ∀ finite Λ0⊂Γ,Ω-*0∈W¯Λ(n), g∈G and a∈(1,∞), for any n large enough, ∀ΩΛ(n)∖Λ0*={Ω*(i),i∈Λ(n)∖Λ0} and ΩΓ∖Λ(n)*={Ω*(i),i∈Γ∖Λ(n)} with ♯Ω*(i)≤κ,
(132)a2exp[-hΛ(n)((gΩ-*0)∨(gΛ(n)∖Λ0ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)]+a2exp[((g-1Ω-*0)∨(gΛ(n)∖Λ0-1ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)-hΛ(n)×((g-1Ω-*0)∨(gΛ(n)∖Λ0-1ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)]≥exp[-hΛ(n)(Ω-*0∨ΩΛ(n)∖Λ0*∣ΩΓ∖Λ(n)*)].In (132), the loop configuration gΛ(n)∖Λ0ΩΛ(n)∖Λ0* is determined by specifying its temporal section {gΛ(n)∖Λ0Ωx,i*(τ+βm),i∈Λ(n)∖Λ0,x∈x*(i),0≤m<kx,i}. That is, we need to specify the sections:
(133)(u(τ+βm;gΛ(n)∖Λ0Ωx,i*),l(τ+βm;gΛ(n)∖Λ0Ωx,i*))
for loops Ωx,i* constituting gΛ(n)∖Λ0ΩΛ(n)∖Λ0*. To this end we set
(134)u(τ+βm;gΛ(n)∖Λ0Ωx,i*)=gj(n)u(τ+βm;Ωx,i*)ifl(τ+βm;gΛ(n)∖Λ0Ωx,i*)=j.
In other words, we apply the action gj(n) to the temporal sections of all loops Ωx,i* located at vertex j at a given time, regardless of position of their initial points (x,i) in Λ(n)∖Λ0.Observe that (130) is deduced from (132) by integrating in dΩΛ(n)∖Λ0* and normalizing by ΞΛ(n)∖Λ0(ΩΓ∖Λ(n)*); see (80) with Λ′=Λ(n). (The Jacobian of the map ΩΛ(n)∖Λ0*↦gΛ(n)∖Λ0ΩΛ(n)∖Λ0* equals 1.)Thus, our aim becomes proving (132). The tuned family gΛ(n)∖Λ0 is composed of individual actions gj(n)∈G:
(135)gΛ(n)∖Λ0={gj(n),j∈Λ(n)∖Λ0}.
Elements gj(n) are powers (multiples, in the additive parlance) of element g∈G figuring in (128)–(132) (resp., of the corresponding vector θ_∈M; cf. (9)) and defined as follows. Let θ_j(n) denote the vector from M corresponding to gj(n), and we select positive integer values r¯(n)=⌈log(1+n)⌉ and set
(136)θ_j(n)=θ_υ(n,j),
where
(137)υ(n,j)={1,d(o,j)≤r¯(n),ϑ(d(j,o)-r¯(n),n-r¯(n)),d(o,j)>r¯(n).
In turn, the function ϑ is chosen to satisfy
(138)ϑ(a,b)=1(a≤0)+1(0<a<b)Q(b)×∫abz(u)du,a,b∈ℝ,
with
(139)Q(b)=∫0b‍ζ(u)du~loglogb,whereζ(u)=1(u≤2)+1(u>2)1ulnu,b>0.Lemma 31.
Givenz,β∈(0,∞) satisfying (22) and a finite set Λ0, there exists a constant C∈(0,∞) such that ∀x0,y0∈M*Λ0, the set of path configurations Ω-*0∈W¯x*0,y*0* with hΛ0(Ω-*0)<+∞ which include trajectories visiting points in Γ∖Λ(r¯(n)) has the ℙ¯x0,y0* measure that does not exceed C/(⌈(1+r(n))⌉)!.Proof of Lemma31.
The condition thathΛ0(Ω-*0)<+∞ implies that the total number of sub trajectories of length β in Ω-*0 does not exceed κ×♯Λ0 which is a fixed value in the context of the lemma. Each such trajectory has a Poisson number of jumps; this produces the factor 1/(⌈(1+r(n))⌉)!.Back to the proof of Theorem29: let gΛ(n)∖Λ0-1 be the collection of the inverse elements:
(140)gΛ(n)∖Λ0-1={gj(n)-1,j∈Λ(n)∖Λ0}.
The vectors corresponding to gj(n)-1 are -θ_j(n)∈M. We will use this specification for gj(n) and gj(n)-1 for j∈Λ(n), or even for j∈Γ, as it agrees with the requirement that gj(n)≡g when j∈Λ0 and gj(n)≡e for j∈Γ∖Λ(n). Accordingly, we will use the notation gΛ(n)={gj(n),j∈Λ(n)}.Observe that the tuned familygΛ(n)∖Λ0 does not change the contribution into the energy functional hΛ(n)∣Γ∖Λ(n) coming from potentials U(1) and U(2): it affects only contributions from potential V.The Taylor formula for functionV, together with the above identification of vectors θ_j(n), gives
(141)|V(gj(n)x,gj′(n)x′)+V(gj(n)-1x,gj′(n)-1x′)-2V(x,x′)|≤C|θ_|2|υ(n,j)-υ(n,j′)|2V¯,x,x′∈M.
Here C∈(0,∞) is a constant, |θ_| stands for the norm of the vector θ_ representing the element g, and the value V¯ is taken from (14).Next, the square|υ(n,j)-υ(n,j′)|2 can be specified as
(142)|υ(n,j)-υ(n,j′)|2={0,ifd(j,o),d(j′,o)≤r¯(n),0,ifd(j,o),d(j′,o)≥n[(d(j′,o)-r¯(n),n-r¯(n))ϑ(d(j,o)-r¯(n),n-r¯(n))-ϑ(d(j′,o)-r¯(n),n-r¯(n))]2,ifr¯(n)<d(j,o),d(j′,o)≤n,ϑ(d(j,o)-r¯(n),n-r¯(n))2,ifr¯(n)<d(j,o)≤n,d(j′,o)∈]r¯(n),n[,ϑ(d(j′,o)-r¯(n),n-r¯(n))2,ifr¯(n)<d(j′,o)≤n,d(j,o)∈]r¯(n),n[.By using convexity of the function exp and (141), ∀a>1,
(143)a2exp[-hΛ(n)(gΛ(n)(Ω-*0∨ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)]+a2exp[-hΛ(n)(gΛ(n)-1(Ω-*0∨ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)]≥aexp[-12hΛ(n)∣Γ∖Λ(n)×(gΛ(n)(Ω-*0∨ΩΛ(n)∖Λ0*),ΩΓ∖Λ(n)*)-12hΛ(n)(gΛ(n)-1(Ω-*0∨ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)]≥aexp[-hΛ(n)(Ω-*0∨ΩΛ(n)∖Λ0*∣ΩΓ∖Λ(n)*)]e-CΥ/2,
where
(144)Υ=Υ(n,g)=βκ2×∑(j,j′)∈Λ(n)×ΓJ(d(j,j′))|υ(n,j)-υ(n,j′)|2.
The next observation is that
(145)Υ≤3βκ2|θ_|2×∑(j,j′)∈Λ(n)×Γ1(d(j,o)≤d(j′,0))J(d(j,j′))×[(j′,o)ϑ(d(j,o)-r¯(n),n-r¯(n))-ϑ(d(j′,o)-r¯(n),n-r¯(n))]2,
where, owing to the triangle inequality, for all j,j′:d(j,o)≤d(j′,o)(146)0≤ϑ(d(j,o)-r¯(n),n-r¯(n))-ϑ(d(j′,o)-r¯(n),n-r¯(n))≤d(j,j′)ζ(d(j,o)-r¯)Q(n-r¯(n)).
This yields
(147)Υ≤βκ23|θ_|2Q(n-r¯(n))2×∑(j,j′)∈Λ(n)×ΓJ(d(j,j′))d(j,j′)2ζ(d(j,0)-r¯(n))2≤3|θ_|2Q(n-r¯(n))2[supj∈Γ∑j′∈ΓJj,j′d(j,j′)2]×∑j∈Λn+r0ζ(d(j,0)-r¯(n))2,
where function ζ is determined in (139).Therefore, it remains to estimate the sum∑j∈Λn+r0ζ(d(j,0)-r¯(n))2. To this end, observe that uζ(u)<1 when u∈(3,∞). The next remark is that the number of sites in the sphere Σn grows linearly with n. Consequently,
(148)∑j∈Λn+r0ζ(d(j,o)-r¯(n))2=∑1≤k≤n+r0ζ(k-r¯(n))×∑j∈Σkζ(k-r¯(n))≤C0∑1≤k≤n+r0ζ(k-r¯(n))≤C1Q(n+r0-r¯(n)),Υ≤CQ(n-r¯(n))⟶∞,asn⟶∞.Therefore, givena>1 for n large enough, the term ae-CΥ/2 in the RHS of (145) becomes >1. Hence,
(149)a2exp[-hΛ(n)(gΛ(n)(Ω-*0∨ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)]+a2exp[-hΛ(n)(gΛ(n)-1(Ω-*0∨ΩΛ(n)∖Λ0*)∣ΩΓ∖Λ(n)*)]≥exp[-hΛ(n)(Ω-*0∨ΩΛ(n)∖Λ0*∣ΩΓ∖Λ(n)*)].
Equation (149) implies that the quantity
(150)qΛ0∣Γ∖Λ(n)(Ω-*0∣ΩΓ∖Λ(n))=∫WΛ(n)∖Λ0dΩΛ(n)∖Λ0*×exp[-hΛ0(Ω-*0∨ΩΛ(n)∖Λ0*∣ΩΓ∖Λ0*)]ΞΛ(n)(ΩΓ∖Λ(n)*),
obeys
(151)limn→∞[qβΛ0∣Γ∖Λ(n)(gΩ-*0∣ΩΓ∖Λ(n)*)+qβΛ0∣Γ∖Λ(n)(g-1Ω-*0∣ΩΓ∖Λ(n)*)]≥2limn→∞qβΛ0∣Γ∖Λ(n)(Ω-*0∣ΩΓ∖Λ(n)*)
uniformly in boundary condition ωΓ∖Λ(n). Integrating (151) dμΓΓ∖Λ(n)(ωΓ∖Λ(n)) yields (132).



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*Source: 637375-2013-09-15.xml*