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 $U$-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 $\sin$ 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 $S_n$, 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 $m$, a positive real number $\lambda$, and a positive integer $k$ such that $(S_n-nm)/n^{\alpha}$ satisfies a moderate deviations principle with speed $n^{1-2k(1-\alpha)}$ and rate function $\lambda x^{2k}/(2k)!$, where $1-1/(2(2k-1)) < \alpha < 1$.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} $S_n/n$ with respect to the associated Gibbs measure, where $S_n/n$ 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 $L^1$-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 $Z_+^2$ spatially homogeneous in the interior, absorbed at the axes, starting from an arbitrary point $(i_0,j_0)$ and with step probabilities drawn on Figure 1 is considered. The trivariate generating function of probabilities that the random walk hits a given point $(i,j)\in Z_+^2$ at a given time $k\geq 0$ is made explicit. Probabilities of absorption at a given time $k$ and at a given axis are found, and their precise asymptotic is derived as the time $k\to\infty$. 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 $Z$, 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