45 research outputs found
Moderate deviations via cumulants
The purpose of the present paper is to establish moderate deviation
principles for a rather general class of random variables fulfilling certain
bounds of the cumulants. We apply a celebrated lemma of the theory of large
deviations probabilities due to Rudzkis, Saulis and Statulevicius. The examples
of random objects we treat include dependency graphs, subgraph-counting
statistics in Erd\H{o}s-R\'enyi random graphs and -statistics. Moreover, we
prove moderate deviation principles for certain statistics appearing in random
matrix theory, namely characteristic polynomials of random unitary matrices as
well as the number of particles in a growing box of random determinantal point
processes like the number of eigenvalues in the GUE or the number of points in
Airy, Bessel, and random point fields.Comment: 24 page
Moderate deviations for random field Curie-Weiss models
The random field Curie-Weiss model is derived from the classical Curie-Weiss
model by replacing the deterministic global magnetic field by random local
magnetic fields. This opens up a new and interestingly rich phase structure. In
this setting, we derive moderate deviations principles for the random total
magnetization , which is the partial sum of (dependent) spins. A typical
result is that under appropriate assumptions on the distribution of the local
external fields there exist a real number , a positive real number
, and a positive integer such that satisfies
a moderate deviations principle with speed and rate
function , where .Comment: 21 page
Large deviations principle for Curie-Weiss models with random fields
In this article we consider an extension of the classical Curie-Weiss model
in which the global and deterministic external magnetic field is replaced by
local and random external fields which interact with each spin of the system.
We prove a Large Deviations Principle for the so-called {\it magnetization per
spin} with respect to the associated Gibbs measure, where is
the scaled partial sum of spins. In particular, we obtain an explicit
expression for the LDP rate function, which enables an extensive study of the
phase diagram in some examples. It is worth mentioning that the model
considered in this article covers, in particular, both the case of i.\,i.\,d.\
random external fields (also known under the name of random field Curie-Weiss
models) and the case of dependent random external fields generated by e.\,g.\
Markov chains or dynamical systems.Comment: 11 page
Large deviations for disordered bosons and multiple orthogonal polynomial ensembles
We prove a large deviations principle for the empirical measures of a class
of biorthogonal and multiple orthogonal polynomial ensembles that includes
biorthogonal Laguerre, Jacobi and Hermite ensembles, the matrix model of Lueck,
Sommers and Zirnbauer for disordered bosons, the Stieltjes-Wigert matrix model
of Chern-Simons theory, and Angelesco ensembles.Comment: 20 page
Mod-Gaussian convergence and its applications for models of statistical mechanics
In this paper we complete our understanding of the role played by the
limiting (or residue) function in the context of mod-Gaussian convergence. The
question about the probabilistic interpretation of such functions was initially
raised by Marc Yor. After recalling our recent result which interprets the
limiting function as a measure of "breaking of symmetry" in the Gaussian
approximation in the framework of general central limit theorems type results,
we introduce the framework of -mod-Gaussian convergence in which the
residue function is obtained as (up to a normalizing factor) the probability
density of some sequences of random variables converging in law after a change
of probability measure. In particular we recover some celebrated results due to
Ellis and Newman on the convergence in law of dependent random variables
arising in statistical mechanics. We complete our results by giving an
alternative approach to the Stein method to obtain the rate of convergence in
the Ellis-Newman convergence theorem and by proving a new local limit theorem.
More generally we illustrate our results with simple models from statistical
mechanics.Comment: 49 pages, 21 figure
Large Deviations for a Non-Centered Wishart Matrix
We investigate an additive perturbation of a complex Wishart random matrix
and prove that a large deviation principle holds for the spectral measures. The
rate function is associated to a vector equilibrium problem coming from
logarithmic potential theory, which in our case is a quadratic map involving
the logarithmic energies, or Voiculescu's entropies, of two measures in the
presence of an external field and an upper constraint. The proof is based on a
two type particles Coulomb gas representation for the eigenvalue distribution,
which gives a new insight on why such variational problems should describe the
limiting spectral distribution. This representation is available because of a
Nikishin structure satisfied by the weights of the multiple orthogonal
polynomials hidden in the background.Comment: 40 page
The Hartree limit of Born's ensemble for the ground state of a bosonic atom or ion
The non-relativistic bosonic ground state is studied for quantum N-body
systems with Coulomb interactions, modeling atoms or ions made of N "bosonic
point electrons" bound to an atomic point nucleus of Z "electron" charges,
treated in Born--Oppenheimer approximation. It is shown that the (negative)
ground state energy E(Z,N) yields the monotonically growing function (E(l N,N)
over N cubed). By adapting an argument of Hogreve, it is shown that its limit
as N to infinity for l > l* is governed by Hartree theory, with the rescaled
bosonic ground state wave function factoring into an infinite product of
identical one-body wave functions determined by the Hartree equation. The proof
resembles the construction of the thermodynamic mean-field limit of the
classical ensembles with thermodynamically unstable interactions, except that
here the ensemble is Born's, with the absolute square of the ground state wave
function as ensemble probability density function, with the Fisher information
functional in the variational principle for Born's ensemble playing the role of
the negative of the Gibbs entropy functional in the free-energy variational
principle for the classical petit-canonical configurational ensemble.Comment: Corrected version. Accepted for publication in Journal of
Mathematical Physic
Passage time from four to two blocks of opinions in the voter model and walks in the quarter plane
A random walk in spatially homogeneous in the interior, absorbed at
the axes, starting from an arbitrary point and with step
probabilities drawn on Figure 1 is considered. The trivariate generating
function of probabilities that the random walk hits a given point at a given time is made explicit. Probabilities of absorption
at a given time and at a given axis are found, and their precise asymptotic
is derived as the time . The equivalence of two typical ways of
conditioning this random walk to never reach the axes is established. The
results are also applied to the analysis of the voter model with two candidates
and initially, in the population , four connected blocks of same opinions.
Then, a citizen changes his mind at a rate proportional to the number of its
neighbors that disagree with him. Namely, the passage from four to two blocks
of opinions is studied.Comment: 11 pages, 1 figur