pinecone/core-2020-05-10-deduplication huggingface dataset:
Epidemic thresholds on scale-free graphs: the interplay between exponent
  and preferential choice

We show for a model of scale-free graphs with biased partner choice that
knowing the exponent for the degree distribution is in general not sufficient
to decide epidemic threshold properties for exponents less than three.We show
that the connectivity between the high degree vertices and therefore the
diameter is the relevant geometric quantity for epidemic threshold
estimations.Absence of epidemic threshold happens precisely when a positive
fraction of the nodes form a cluster of bounded diameter.Comment: 10 page
Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions I

This is the first in a series of works devoted to small non-selfadjoint
perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2.
In the present work we treat the case when the classical flow of the
unperturbed part is periodic and the strength $\epsilon$ of the perturbation is
$\gg h$ (or sometimes only $\gg h^2$) and bounded from above by $h^{\delta}$
for some $\delta>0$. We get a complete asymptotic description of all
eigenvalues in certain rectangles $[-1/C, 1/C]+ i\epsilon [F_0-1/C,F_0+1/C]$.Comment: 81 page
Outgoing radiation from an isolated collisionless plasma

The asymptotic properties at future null infinity of the solutions of the
relativistic Vlasov-Maxwell system whose global existence for small data has
been established by the author in a previous work are investigated. These
solutions describe a collisionless plasma isolated from incoming radiation. It
is shown that a non-negative quantity associated to the plasma decreases as a
consequence of the dissipation of energy in form of outgoing radiation. This
quantity represents the analogue of the Bondi mass in general relativity.Comment: 13 pages; version in press. This paper continues the analysis started
  in math-ph/021101
Proof of the Ergodic Hypothesis for Typical Hard Ball Systems

We consider the system of $N$ ($\ge2$) hard balls with masses $m_1,...,m_N$
and radius $r$ in the flat torus $\Bbb T_L^\nu=\Bbb R^\nu/L\cdot\Bbb Z^\nu$ of
size $L$, $\nu\ge3$. We prove the ergodicity (actually, the Bernoulli mixing
property) of such systems for almost every selection $(m_1,...,m_N; L)$ of the
outer geometric parameters. This theorem complements my earlier result that
proved the same, almost sure ergodicity for the case $\nu=2$. The method of
that proof was primarily dynamical-geometric, whereas the present approach is
inherently algebraic.Comment: 31 pages, no figure
Conformal transformations and the SLE partition function martingale

We present an implementation in conformal field theory (CFT) of local finite
conformal transformations fixing a point. We give explicit constructions when
the fixed point is either the origin or the point at infinity. Both cases
involve the exponentiation of a Borel subalgebra of the Virasoro algebra. We
use this to build coherent state representations and to derive a close analog
of Wick's theorem for the Virasoro algebra. This allows to compute the
conformal partition function in non trivial geometries obtained by removal of
hulls from the upper half plane. This is then applied to stochastic Loewner
evolutions (SLE). We give a rigorous derivation of the equations, obtained
previously by the authors, that connect the stochastic Loewner equation to the
representation theory of the Virasoro algebra. We give a new proof that this
construction enumerates all polynomial SLE martingales. When one of the hulls
removed from the upper half plane is the SLE hull, we show that the partition
function is a famous local martingale known to probabilists, thereby
unravelling its CFT origin.Comment: 41 pages, 4 figure
Uniform Lieb-Thirring inequality for the three dimensional Pauli
  operator with a strong non-homogeneous magnetic field

The Pauli operator describes the energy of a nonrelativistic quantum particle
with spin 1/2 in a magnetic field and an external potential. A new
Lieb-Thirring type inequality on the sum of the negative eigenvalues is
presented. The main feature compared to earlier results is that in the large
field regime the present estimate grows with the optimal (first) power of the
strength of the magnetic field. As a byproduct of the method, we also obtain an
optimal upper bound on the pointwise density of zero energy eigenfunctions of
the Dirac operator. The main technical tools are:
  (i) a new localization scheme for the square of the resolvent of a general
class of second order elliptic operators;
  (ii) a geometric construction of a Dirac operator with a constant magnetic
field that approximates the original Dirac operator in a tubular neighborhood
of a fixed field line. The errors may depend on the regularity of the magnetic
field but they are uniform in the field strength.Comment: latex file. Revised final version: typos corrected, the definition of
  the lengthscale simplified, references added/update
Rotating Singular Perturbations of the Laplacian

We study a system of a quantum particle interacting with a singular
time-dependent uniformly rotating potential in 2 and 3 dimensions: in
particular we consider an interaction with support on a point (rotating point
interaction) and on a set of codimension 1 (rotating blade). We prove the
existence of the Hamiltonians of such systems as suitable self-adjoint
operators and we give an explicit expression for their unitary semigroups.
Moreover we analyze the asymptotic limit of large angular velocity and we prove
strong convergence of the time-dependent propagator to some one-parameter
unitary group as (\omega \to \infty).Comment: Minor changes, to appear in Ann. H. Poincare', 35 pages, LaTe
Scaling algebras for charged fields and short-distance analysis for
  localizable and topological charges

The method of scaling algebras, which has been introduced earlier as a means
for analyzing the short-distance behaviour of quantum field theories in the
setting of the model-independent, operator-algebraic approach, is extended to
the case of fields carrying superselection charges. In doing so, consideration
will be given to strictly localizable charges ("DHR-type" superselection
charges) as well as to charges which can only be localized in regions extending
to spacelike infinity ("BF-type" superselection charges). A criterion for the
preservance of superselection charges in the short-distance scaling limit is
proposed. Consequences of this preservance of superselection charges are
studied. The conjugate charge of a preserved charge is also preserved, and for
charges of DHR-type, the preservance of all charges of a quantum field theory
in the scaling limit leads to equivalence of local and global intertwiners
between superselection sectors.Comment: Latex 2e, 57 pages. Supersedes hep-th/030114
Asymptotics of solutions of the Einstein equations with positive
  cosmological constant

A positive cosmological constant simplifies the asymptotics of forever
expanding cosmological solutions of the Einstein equations. In this paper a
general mathematical analysis on the level of formal power series is carried
out for vacuum spacetimes of any dimension and perfect fluid spacetimes with
linear equation of state in spacetime dimension four. For equations of state
stiffer than radiation evidence for development of large gradients, analogous
to spikes in Gowdy spacetimes, is found. It is shown that any vacuum solution
satisfying minimal asymptotic conditions has a full asymptotic expansion given
by the formal series. In four spacetime dimensions, and for spatially
homogeneous spacetimes of any dimension, these minimal conditions can be
derived for appropriate initial data. Using Fuchsian methods the existence of
vacuum spacetimes with the given formal asymptotics depending on the maximal
number of free functions is shown without symmetry assumptions.Comment: 23 page
Modular Nuclearity and Localization

Within the algebraic setting of quantum field theory, a condition is given
which implies that the intersection of algebras generated by field operators
localized in wedge--shaped regions of two--dimensional Minkowski space is
non--trivial; in particular, there exist compactly localized operators in such
theories which can be interpreted as local observables. The condition is based
on spectral (nuclearity) properties of the modular operators affiliated with
wedge algebras and the vacuum state and is of interest in the algebraic
approach to the formfactor program, initiated by Schroer. It is illustrated
here in a simple class of examples.Comment: 14 pages, no figure
On the Vacuum Polarization Density Caused by an External Field

We consider an external potential, $-\lambda \phi$, due to one or more
nuclei. Following the Dirac picture such a potential polarizes the vacuum. The
polarization density as derived in physics literature, after a well known
renormalization procedure, depends decisively on the strength of $\lambda$. For
small $\lambda$, more precisely as long as the lowest eigenvalue,
$e_1(\lambda)$, of the corresponding Dirac operator stays in the gap of the
essential spectrum, the integral over the density vanishes. In other words the
vacuum stays neutral. But as soon as $e_1(\lambda)$ dives into the lower
continuum the vacuum gets spontaneously charged with charge $ 2e$. Global
charge conservation implies that two positrons were emitted out of the vacuum,
this is, a large enough external potential can produce electron-positron pairs.
  We give a rigorous proof of that phenomenon.Comment: proofs correcte
Perturbative test of single parameter scaling for 1D random media

Products of random matrices associated to one-dimensional random media
satisfy a central limit theorem assuring convergence to a gaussian centered at
the Lyapunov exponent. The hypothesis of single parameter scaling states that
its variance is equal to the Lyapunov exponent. We settle discussions about its
validity for a wide class of models by proving that, away from anomalies,
single parameter scaling holds to lowest order perturbation theory in the
disorder strength. However, it is generically violated at higher order. This is
explicitely exhibited for the Anderson model.Comment: minor corrections to previous version, to appear in Annales H.
  Poincar
Order by disorder, without order, in a two-dimensional spin system with
  O(2) symmetry

We present a rigorous proof of an ordering transition for a two-component
two-dimensional antiferromagnet with nearest and next-nearest neighbor
interactions. The low-temperature phase contains two states distinguished by
local order among columns or, respectively, rows. Overall, there is no magnetic
order in accord with the classic Mermin-Wagner theorem. The method of proof
employs a rigorous version of "order by disorder," whereby a high degeneracy
among the ground states is lifted according to the differences in their
associated spin-wave spectra.Comment: 22 pages, 1 eps fi
Some connections between Dirac-Fock and Electron-Positron Hartree-Fock

We study the ground state solutions of the Dirac-Fock model in the case of
weak electronic repulsion, using bifurcation theory. They are solutions of a
min-max problem. Then we investigate a max-min problem coming from the
electron-positron field theory of Bach-Barbaroux-Helffer-Siedentop. We show
that given a radially symmetric nuclear charge, the ground state of Dirac-Fock
solves this max-min problem for certain numbers of electrons. But we also
exhibit a situation in which the max-min level does not correspond to a
solution of the Dirac-Fock equations together with its associated
self-consistent projector
KIDs are non-generic

We prove that generic solutions of the vacuum constraint Einstein equations
do not possess any global or local space-time Killing vectors, on an
asymptotically flat Cauchy surface, or on a compact Cauchy surface with mean
curvature close to a constant, or for CMC asymptotically hyperbolic initial
data sets. More generally, we show that non-existence of global symmetries
implies, generically, non-existence of local ones. As part of the argument, we
prove that generic metrics do not possess any local or global conformal Killing
vectors.Comment: latex2e, 38 pages in A4 now, minor corrections throughou
Product formula related to quantum Zeno dynamics

We prove a product formula which involves the unitary group generated by a
semibounded self-adjoint operator and an orthogonal projection $P$ on a
separable Hilbert space $\HH$, with the convergence in
$L^2_\mathrm{loc}(\mathbb{R};\HH)$. It gives a partial answer to the question
about existence of the limit which describes quantum Zeno dynamics in the
subspace \hbox{$\mathrm{Ran} P$}. The convergence in $\HH$ is demonstrated in
the case of a finite-dimensional $P$. The main result is illustrated in the
example where the projection corresponds to a domain in $\mathbb{R}^d$ and the
unitary group is the free Schr\"odinger evolution.Comment: LaTeX 2e, 24 pages, with substantial modifications, to appear in Ann.
  H. Poincar
Precise coupling terms in adiabatic quantum evolution

It is known that for multi-level time-dependent quantum systems one can
construct superadiabatic representations in which the coupling between
separated levels is exponentially small in the adiabatic limit. For a family of
two-state systems with real-symmetric Hamiltonian we construct such a
superadiabatic representation and explicitly determine the asymptotic behavior
of the exponentially small coupling term. First order perturbation theory in
the superadiabatic representation then allows us to describe the
time-development of exponentially small adiabatic transitions. The latter
result rigorously confirms the predictions of Sir Michael Berry for our family
of Hamiltonians and slightly generalizes a recent mathematical result of George
Hagedorn and Alain Joye.Comment: 24 page
Absence of continuous spectral types for certain nonstationary random
  models

We consider continuum random Schr\"odinger operators of the type $H_{\omega}
= -\Delta + V_0 + V_{\omega}$ with a deterministic background potential $V_0$.
We establish criteria for the absence of continuous and absolutely continuous
spectrum, respectively, outside the spectrum of $-\Delta +V_0$. The models we
treat include random surface potentials as well as sparse or slowly decaying
random potentials. In particular, we establish absence of absolutely continuous
surface spectrum for random potentials supported near a one-dimensional surface
(``random tube'') in arbitrary dimension.Comment: 14 pages, 2 figure
Spectrum of the Magnetic Schrodinger Operator in a Waveguide with
  Combined Boundary Conditions

We consider the magnetic Schrodinger operator in a two-dimensional strip. On
the boundary of the strip the Dirichlet boundary condition is imposed except
for a fixed segment (window), where it switches to magnetic Neumann boundary
condition (see Section 2, Eq. (2.2) for the definition of this boundary
condition}. We deal with a smooth compactly supported field as well as with the
Aharonov-Bohm field. We give an estimate on the maximal length of the window,
for which the discrete spectrum of the considered operator will be empty. In
the case of a compactly supported field we also give a sufficient condition for
the presence of eigenvalues below the essential spectrum
The Hopf algebra of rooted trees in Epstein-Glaser renormalization

We show how the Hopf algebra of rooted trees encodes the combinatorics of
Epstein-Glaser renormalization and coordinate space renormalization in general.
In particular we prove that the Epstein-Glaser time-ordered products can be
obtained from the Hopf algebra by suitable Feynman rules, mapping trees to
operator-valued distributions. Twisting the antipode with a renormalization map
formally solves the Epstein-Glaser recursion and provides local counterterms
due to the Hochschild 1-closedness of the grafting operator $B_+$.Comment: 19p, minor corrections and improvements. To appear in AH
Renormalization of the 2-point function of the Hubbard model at
  half-filling

We prove that the two dimensional Hubbard model at finite temperature T and
half-filling is analytic in the coupling constant in a radius at least $c/(\log
T)^2$. We also study the self-energy through a new two-particle irreducible
expansion and prove that this model is not a Fermi liquid, but a Luttinger
liquid with logarithmic corrections. The techniques used are borrowed from
constructive field theory so the result is mathematically rigorous and
completely non-perturbative.
  Together with earlier results on the existence of two dimensional Fermi
liquids, this new result proves that the nature of interacting Fermi systems in
two dimensions depends on the shape of the Fermi surface.Comment: 45 pages, 28 figure
Phase Space Reduction of Star Products on Cotangent Bundles

In this paper we construct star products on Marsden-Weinstein reduced spaces
in case both the original phase space and the reduced phase space are
(symplectomorphic to) cotangent bundles. Under the assumption that the original
cotangent bundle $T^*Q$ carries a symplectique structure of form
$\omega_{B_0}=\omega_0 + \pi^*B_0$ with $B_0$ a closed two-form on $Q$, is
equipped by the cotangent lift of a proper and free Lie group action on $Q$ and
by an invariant star product that admits a $G$-equivariant quantum momentum
map, we show that the reduced phase space inherits from $T^*Q$ a star product.
Moreover, we provide a concrete description of the resulting star product in
terms of the initial star product on $T^*Q$ and prove that our reduction scheme
is independent of the characteristic class of the initial star product. Unlike
other existing reduction schemes we are thus able to reduce not only strongly
invariant star products. Furthermore in this article, we establish a relation
between the characteristic class of the original star product and the
characteristic class of the reduced star product and provide a classification
up to $G$-equivalence of those star products on $(T^*Q,\omega_{B_0})$, which
are invariant with respect to a lifted Lie group action. Finally, we
investigate the question under which circumstances `quantization commutes with
reduction' and show that in our examples non-trivial restrictions arise
Geometric modular action and spontaneous symmetry breaking

We study spontaneous symmetry breaking for field algebras on Minkowski space
in the presence of a condition of geometric modular action (CGMA) proposed
earlier as a selection criterion for vacuum states on general space-times. We
show that any internal symmetry group must commute with the representation of
the Poincare group (whose existence is assured by the CGMA) and each
translation-invariant vector is also Poincare invariant. The subspace of these
vectors can be centrally decomposed into pure invariant states and the CGMA
holds in the resulting sectors. As positivity of the energy is not assumed,
similar results may be expected to hold for other space--times.Comment: Dedicated to the memory of Siegfried Schlieder. 17 pages, no figures.
  Revised version: simplified arguments and improved results; as to appear in
  Annales H. Poincar
Semiclassical propagation of coherent states with spin-orbit interaction

We study semiclassical approximations to the time evolution of coherent
states for general spin-orbit coupling problems in two different semiclassical
scenarios: The limit \hbar to zero is first taken with fixed spin quantum
number s and then with \hbar*s held constant. In these two cases different
classical spin-orbit dynamics emerge. We prove that a coherent state propagated
with a suitable classical dynamics approximates the quantum time evolution up
to an error of size \sqrt{\hbar} and identify an Ehrenfest time scale.
Subsequently an improvement of the semiclassical error to an arbitray order
\hbar^{N/2} is achieved by a suitable deformation of the state that is
propagated classically
Quantum backreaction (Casimir) effect I. What are admissible
  idealizations?

Casimir effect, in a broad interpretation which we adopt here, consists in a
backreaction of a quantum system to adiabatically changing external conditions.
Although the system is usually taken to be a quantum field, we show that this
restriction rather blurs than helps to clarify the statement of the problem. We
discuss the problem from the point of view of algebraic structure of quantum
theory, which is most appropriate in this context. The system in question may
be any quantum system, among others both finite as infinite dimensional
canonical systems are allowed. A simple finite-dimensional model is discussed.
We identify precisely the source of difficulties and infinities in most of
traditional treatments of the problem for infinite dimensional systems (such as
quantum fields), which is incompatibility of algebras of observables or their
representations. We formulate conditions on model idealizations which are
acceptable for the discussion of the adiabatic backreaction problem. In the
case of quantum field models in that class we find that the normal ordered
energy density is a well defined distribution, yielding global energy in the
limit of a unit test function. Although we see the "zero point" expressions as
inappropriate, we show how they can arise in the quantum field theory context
as a result of uncontrollable manipulations.Comment: 40 pages, AMS-LaTeX; to appear in Ann. H. Poincar
Spin, Statistics, and Reflections, I. Rotation Invariance

The universal covering of SO(3) is modelled as a reflection group G_R in a
representation independent fashion. For relativistic quantum fields, the Unruh
effect of vacuum states is known to imply an intrinsic form of reflection
symmetry, which is referred to as "modular P_1CT-symmetry (Bisognano, Wichmann,
1975, 1976, and Guido, Longo, [funct-an/9406005]). This symmetry is used to
construct a representation of G_R by pairs of modular P_1CT-operators. The
representation thus obtained satisfies Pauli's spin-statistics relation.Comment: Accepted for publication in Ann. H. Poincare, (annoying) misprints
  correcte
Spin-Glass Stochastic Stability: a Rigorous Proof

We prove the property of stochastic stability previously introduced as a
consequence of the (unproved) continuity hypothesis in the temperature of the
spin-glass quenched state. We show that stochastic stability holds in
beta-average for both the Sherrington-Kirkpatrick model in terms of the square
of the overlap function and for the Edwards-Anderson model in terms of the bond
overlap. We show that the volume rate at which the property is reached in the
thermodynamic limit is V^{-1}. As a byproduct we show that the stochastic
stability identities coincide with those obtained with a different method by
Ghirlanda and Guerra when applyed to the thermal fluctuations only.Comment: 12 pages, revised versio
Determination of Non-Adiabatic Scattering Wave Functions in a
  Born-Oppenheimer Model

We study non--adiabatic transitions in scattering theory for the time
dependent molecular Schroedinger equation in the Born--Oppenheimer limit. We
assume the electron Hamiltonian has finitely many levels and consider the
propagation of coherent states with high enough total energy. When two of the
electronic levels are isolated from the rest of the electron Hamiltonian's
spectrum and display an avoided crossing, we compute the component of the
nuclear wave function associated with the non--adiabatic transition that is
generated by propagation through the avoided crossing. This component is shown
to be exponentially small in the square of the Born--Oppenheimer parameter, due
to the Landau-Zener mechanism. It propagates asymptotically as a free Gaussian
in the nuclear variables, and its momentum is shifted. The total transition
probability for this transition and the momentum shift are both larger than
what one would expect from a naive approximation and energy conservation