The Model

Number-Rigidity

Equivalence of ensembles

Number-Rigidity and β -Circular Riesz
GasDavid Dereudre(joint work with Thibaut Vasseur)
Laboratoire de Mathématiques Paul Painlevé,
University of Lille, FranceWorkshop on Random Point Processes in Statistical physics,
June-July 2022, Berlin

The Model

Number-Rigidity

Equivalence of ensembles

Introduction

Gibbs point process on Rd interacting with the Riesz pairpotential
g(x) =1|x|sd−1<s<dg is non-integrable at innity, ∇g is integrable.

canonical ensemble with constant density ρ > 0 and inverse
temperature β > 0.
periodic boundary condition
number-rigidity and equivalence of ensembles

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Number-Rigidity

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Equivalence of ensembles

Equivalence of ensembles

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Equivalence of ensembles

The Riesz energy with backgroundγ = {x1 , . . . , xn } included Λn = [−n1/d /2, n1/d /2]dZ Z
X
1
H(γ) =g(x − y) =g(x − y)γ(dx)γ(dy).
2
Rd \Diag{x,y}∈γ

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Number-Rigidity

Equivalence of ensembles

The Riesz energy with backgroundγ = {x1 , . . . , xn } included Λn = [−n1/d /2, n1/d /2]dZ ZX1H(γ) =g(x − y) =g(x − y)γ(dx)γ(dy).
2Rd \Diag{x,y}∈γWith the background on ΛnH̃n (γ) =12Z Z

Diagg(x − y)(γ(dx) − dx)(γ(dy) − dy).

Λn \The energy H̃n (γ) is of order n (the volume).

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Number-Rigidity

Equivalence of ensembles

The periodic Riesz energy

For k ≥ 1, γ k is the concatenation of (2k + 1) copies of γ in thetranslations of Λn . It is a conguration in Λ(2k+1)d n .

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Number-Rigidity

Equivalence of ensembles

The periodic Riesz energy

For k ≥ 1, γ k is the concatenation of (2k + 1) copies of γ in thetranslations of Λn . It is a conguration in Λ(2k+1)d n .
Proposition
limH̃Λ(2k+1)d n (γ k )(2k + 1)dk→∞

with gn (x) =P=k∈Zd (g(x + knX

gn (x − y) + nεn{x,y}∈γ1/d ) − 1nR

Λn g(y + knFor all x ∈ Λn , |gn (x) − g(x)| ≤ Cn−s/d .

1/d )dy).

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Number-Rigidity

Equivalence of ensembles

The periodic Riesz energy

For k ≥ 1, γ k is the concatenation of (2k + 1) copies of γ in thetranslations of Λn . It is a conguration in Λ(2k+1)d n .
PropositionlimH̃Λ(2k+1)d n (γ k )(2k + 1)dk→∞

with gn (x) =PX=

gn (x − y) + nεn{x,y}∈γk∈Zd (g(x + kn1/d ) − 1nR

Λn g(y + knFor all x ∈ Λn , |gn (x) − g(x)| ≤ Cn−s/d .
Denition
The periodic Riesz energy of γ in Λn is dened byHn (γ) =X{x,y}⊂γgn (x − y).

1/d )dy).

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Number-Rigidity

Equivalence of ensembles

Properties of gn

gn (x) =Xk∈Zd

1/d(g(x + kn1)−nZg(y + kn1/d )dy).

Λn

Proposition(Stability) There exists a constant A ≥ 0 such that for pointconguration γ ∈ Λn such that |γ| = n, we have
Hn (γ) ≥ −An.
(Shift invariance) For every u ∈ Λn and every conguration
γ in Λn we have Hn (τun (γ)) = Hn (γ).
(Approximation) There exists a constant c > 0 such that forevery point x ∈ Λn we have|gn (x) − g(x)| ≤ cn−s/dThe Model

Number-Rigidity

Equivalence of ensembles

The canonical ensembleBinΛ,n is the distribution of n independent points uniformlydistributed in Λ.

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Number-Rigidity

Equivalence of ensembles

The canonical ensembleBinΛ,n is the distribution of n independent points uniformlydistributed in Λ.
Denition
The canonical Gibbs measure in Λn with inverse temperature
β > 0 is1Pnβ = β e−βHn BinΛn ,n .
Zn

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Number-Rigidity

Equivalence of ensembles

The canonical ensembleBinΛ,n is the distribution of n independent points uniformly
distributed in Λ.
Denition
The canonical Gibbs measure in Λn with inverse temperature
β > 0 is1
Pnβ = β e−βHn BinΛn ,n .
Zn

Theorem
The sequence (Pnβ )n≥1 admits accumulation points for the local
convergence topology. They are called β -circular Riesz gases.
Uniqueness or non-uniqueness of accumulation points isunknown.

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Equivalence of ensembles

Main arguments of the proofThe energy is stable : For any γ such that #(γ) = nHn (γ) ≥ −An.

The partition function : There exists 0 < aβ < bβ < +∞anβ ≤ Znβ ≤ bnβ .

The relative entropy is uniformly boundedI(Pnβ |πΛn )/|Λn | ≤ C.
Pnβ is stationary on the torus Λn .

The Model

Number-Rigidity

Equivalence of ensembles

Connections with other models

Hardin, Sa and Simanek (2014) : Periodic energy of acrystalPhysicists : Periodic jellium (s = d − 2)
Leblé-Serfaty (2017) : LDP with conning potential
Valko,Virag (2009), Killip-Stoiciu (2009), Nakano (2014)beta-circular ensembles and the Sine-β process (s = 0,d = 1)
Boursier (2022) : Riesz gas on the circle (0 < s < 1, d = 1)
Lewin (2022) : Survey on Riesz gas

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Number-Rigidity

Number-Rigidity

Equivalence of ensembles

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Number-Rigidity

Equivalence of ensembles

Number-RigidityDenition (Ghosh-Peres 2017)
A point process Γ in Rd is said number-rigid if for any boundedset Λ ⊂ Rd there exists a function FΛ such that almost surely#ΓΛ = FΛ (ΓΛc ).

Are the β -circular gases number-rigid ?

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Number-Rigidity

Equivalence of ensembles

Number-RigidityDenition (Ghosh-Peres 2017)
A point process Γ in Rd is said number-rigid if for any boundedset Λ ⊂ Rd there exists a function FΛ such that almost surely#ΓΛ = FΛ (ΓΛc ).

Are the β -circular gases number-rigid ?
Previous works for Gibbs point process :s > d summable potential : Non number-rigidity (grandcanonical DLR equations)s = 0, d = 2 and β = 2 : Number-Rigidity (DPP structure+ linear statistics), Ghosh-Lebowitz 2017s = 0, d = 1 and β > 0 : Number-Rigidity (canonical DLR
equations or linear statistics), D.-Leblé-Hardy-Maïda 2019
or Chhaibi-Najnudel 2018.

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Number-Rigidity

Equivalence of ensembles

One point deletionDenition (Holroyd-Soo 2013)
A point process Γ in Rd is said one-point deletion if for any
random variate X ⊂ Γ the distribution of Γ\X is absolutely
continuous with respect to Γ.
"Non number-rigidity" and "One point deletion" are almost
equivalent.

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Number-Rigidity

Equivalence of ensembles

One point deletionDenition (Holroyd-Soo 2013)A point process Γ in Rd is said one-point deletion if for any
random variate X ⊂ Γ the distribution of Γ\X is absolutely
continuous with respect to Γ.
"Non number-rigidity" and "One point deletion" are almostequivalent.
Heuristically, for Gibbs point processes and if X is "typical"PΓ∼ e−βh(X,Γ\X) .
PΓ\XThe one point deletion property requires a good denition forh(X, Γ\X).

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Number-Rigidity

Equivalence of ensembles

The energy of a pointLet x ∈ Rd and γ an innite conguration (x ∈/ γ). Threecandidates for h(x, γ) :1h1 (x, γ) ==|x − y|sy∈γ
XZ1γ(dy)|x − y|sZh2 (x, γ) = limn→∞Zh3 (x, γ) = limn→∞1(γ(dy) − dy)sΛn |x − y|1γ(dy) − Cn (#γΛn , γΛcn )s
Λn |x − y|The Model

Number-Rigidity

Equivalence of ensembles

The main result

Theorem
For any β > 0, there exists a β -circular Riesz gas P⋆β which is
not number-rigid. P⋆β is also one-point deletion.
P⋆β = limk→∞ Pnβk for a subsequence (nk ).
We believe that all β -circular Riesz gas are not number-rigid.

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Number-Rigidity

Equivalence of ensembles

The main result

Theorem
For any β > 0, there exists a β -circular Riesz gas P⋆β which is
not number-rigid. P⋆β is also one-point deletion.
P⋆β = limk→∞ Pnβk for a subsequence (nk ).
We believe that all β -circular Riesz gas are not number-rigid.

Corollary
For any bounded Λ and k ≥ 0 then for all P⋆β -a.s. γ ,P⋆β (NΛ = k|γΛc ) > 0.

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Number-Rigidity

Equivalence of ensembles

Main ingredient of the proof

Proposition
For any β > 0, there exists a constant K > 0 and ansubsequence (nk )k≥1 such that for all k ≥ 1
Z|hnk (0, γ)|Pnβk (dγ) ≤ K,

wherehn (x, γ) =X

gn (x − y),y∈γgn (x) =Xk∈Zd1/d(g(x + kn1)−nZ
Λng(y + kn1/d )dy).

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Equivalence of ensembles

Equivalence of ensembles

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Number-Rigidity

Equivalence of ensembles

General principle

The density of particles ρ > 0 isprescribed. In the thermodynamic limit (Λn → ∞) the numberof particles is xed equal to ρ|Λn |.

Canonical ensembles :The Model

Number-Rigidity

Equivalence of ensembles

General principle

The density of particles ρ > 0 isprescribed. In the thermodynamic limit (Λn → ∞) the numberof particles is xed equal to ρ|Λn |.
Grand canonical ensembles : The activity z > 0 (or thechemical potential µ) is prescribed (z = e−βµ ). During thethermodynamic limit (Λn → ∞) the number of particles israndom. The Gibbs process is absolutely continuous withrespect to the Poisson point process with intensity z > 0.

Canonical ensembles :The Model

Number-Rigidity

Equivalence of ensembles

General principle

The density of particles ρ > 0 isprescribed. In the thermodynamic limit (Λn → ∞) the numberof particles is xed equal to ρ|Λn |.
Grand canonical ensembles : The activity z > 0 (or thechemical potential µ) is prescribed (z = e−βµ ). During thethermodynamic limit (Λn → ∞) the number of particles israndom. The Gibbs process is absolutely continuous withrespect to the Poisson point process with intensity z > 0.

Canonical ensembles :Denition (Equivalence of ensembles)
The canonical ensembles and the grand canonical ensembles arethe same. There exist functions ρ 7→ zρ and z 7→ zρ .
The equivalence of ensembles is proved for a large class ofsummable pairwise potentials (Ruelle 70, Georgii 94, Vasseur2012), including the Riesz potential for s > d.

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Number-Rigidity

Equivalence of ensembles

Equivalence of ensembles with the DLR formalism

A canonical ensemble P satises the canonical DLR(Dobrushin-Lanford-Ruelle) equations :P (dγΛ |#γΛ = k, γΛc ) =1ZΛβ (k, γΛc )e−βH(γΛ |γΛc ) BinΛ,k (dγΛ ).

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Number-Rigidity

Equivalence of ensembles

Equivalence of ensembles with the DLR formalism

A canonical ensemble P satises the canonical DLR(Dobrushin-Lanford-Ruelle) equations :P (dγΛ |#γΛ = k, γΛc ) =1ZΛβ (k, γΛc )e−βH(γΛ |γΛc ) BinΛ,k (dγΛ ).

A grand canonical ensemble P satises the grand canonical
DLR equations :P (dγΛ |γΛc ) =1ZΛβ (γΛc )ze−βH(γΛ |γΛc ) πΛ
(dγΛ ).

The ModelNumber-Rigidity

Equivalence of ensembles

Equivalence of ensembles with the DLR formalism

A canonical ensemble P satises the canonical DLR(Dobrushin-Lanford-Ruelle) equations :P (dγΛ |#γΛ = k, γΛc ) =1ZΛβ (k, γΛc )e−βH(γΛ |γΛc ) BinΛ,k (dγΛ ).

A grand canonical ensemble P satises the grand canonical
DLR equations :P (dγΛ |γΛc ) =1ZΛβ (γΛc )ze−βH(γΛ |γΛc ) πΛ
(dγΛ ).

Denition (Equivalence of ensembles)
If P satises the canonical DLR equations then P satises thegrand canonical DLR equations.

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Equivalence of ensembles

Canonical DLR equations for β -circular Riesz gas

The energy to move a particle from 0 to x in γ isM (x|γ) =X

g(x − y) − g(y).

y∈γTheorem (Canonical DLR equations)
Let Pβ be a β -Circular Riesz gas, Λ be a bounded Borel subset ofRd . Then for P β a.e. γ
P β (dγΛ |#γΛ = k, γΛc ) =where H(γΛ |γΛc ) =P1ZΛβ (k, γΛc )e−βH(γΛ |γΛc ) BinΛ,k (dγΛ ).

{x,y}⊂γΛ g(x − y) +x∈γΛ M (x|γΛc ).

P

Similar proof as D.,Leblé,Hardy and Maïda for the Sine-β
process.

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Equivalence of ensembles

βGrand canonical DLR equations for P⋆

Based on the one-point deletion property of P⋆βTheorem (Grand canonical DLR equations)Let Λ be a bounded Borel subset of Rd . Then for P⋆β a.e. γP⋆β (dγΛ |γΛc ) =1ZΛβ (γΛc )e−βH(γΛ |γΛc ) πΛ (dγΛ ).

where γΛ = {x1 , x2 , . . . , xk } andH(γΛ |γΛc ) = h(x1 , γΛc )+h(x2 , x1 ∪γΛc )+. . .+h(x2 , x1 ∪. . .∪xk−1 ∪γΛc )h(x, γ) = lim n→∞Xy∈γΛng(x − y) − Cn (#γΛn , γΛcn ) .

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Equivalence of ensembles

Integral compensatorh(x, γ) = lim n→∞Xg(x − y) − Cn (#γΛn , γΛcn ) .

y∈γΛn

We believe that the integral compensator works
ZCn (#γΛn , γΛcn ) =g(y)dyΛnProposition
If P β is hyperuniform with Var(NΛ ) ≤ C|Λ|s/d−ε then the grandcanonical DLR equations hold withh(x, γ) = lim n→∞

XZg(x − y) −y∈γΛn

It is the case for d = 1, Boursier 2022.

g(y)dy  .
ΛnThe Model

Number-Rigidity

Equivalence of ensembles

Summary of the talkWe dene innite volume Riesz gases (d − 1 < s < d) in Rd
at inverse β > 0 with periodic boundary conditions.
At least one of them P⋆β is not number Rigid.
P⋆β satises canonical and grand canonical DLR equations.
The energy of a point x in γ exists
h(x, γ) = lim n→∞Xg(x − y) − Cn (#γΛn , γΛcn ) .

y∈γΛn

If d = 1, P⋆β is hyperuniform and so
h(x, γ) = lim n→∞Xy∈γΛn

Zg(x − y) −g(y)dy  .
ΛnWe believe that the same hold for all d ≥ 2.

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Open questions

Hyperuniformity and integral compensator for d ≥ 2.
DLR equations for s ≤ d − 1 ?
Does the Number-Rigidity property appear at s = d − 1,s = 0 ? (true for d = 1)
Is it really possible to have Number-Rigidity for large d ?