A large deviation principle for empirical measures on Polish spaces: Application to singular Gibbs measures on manifolds

TL;DR

This paper establishes a large deviation principle for empirical measures under Gibbs measures on Polish spaces, with applications to Coulomb gases, Fekete points, and Gibbs measures on manifolds, using a general Laplace principle approach.

Contribution

It introduces a broad Laplace principle framework for Gibbs measures on Polish spaces, extending large deviation results to singular and manifold settings.

Findings

Proves a large deviation principle for Gibbs point processes.

Applies the results to Coulomb gases and Fekete points.

Provides a general approach inspired by Dupuis et al. for singular Gibbs measures.

Abstract

We prove a large deviation principle for a sequence of point processes defined by Gibbs probability measures on a Polish space. This is obtained as a consequence of a more general Laplace principle for the non-normalized Gibbs measures. We consider three main applications: Conditional Gibbs measures on compact spaces, Coulomb gases on compact Riemannian manifolds and the usual Gibbs measures in the Euclidean space. Finally, we study the generalization of Fekete points and prove a deterministic version of the Laplace principle known as $\Gamma$-convergence. The approach is partly inspired by the works of Dupuis and co-authors. It is remarkably natural and general compared to the usual strategies for singular Gibbs measures.

Equations

$$W_n:M^n\to (-\infty ,\infty ]$$

$$d{\gamma }_n=e^{-n{\beta }_n{W}_n}d{\pi }^{{\otimes }_n}.$$

$${\tilde{W}}_n(\mu )=\{\begin{array}{ll}{W}_n(x_1,...,x_n)& \text{mbox}if\mu \text{mbox}isatomicwith\mu =\frac{1}{n}{\sum }_{i=1}^n{\delta }_{x_i}\\ \infty & \text{mbox}otherwise.\end{array}$$

$$W_n\ge C\text{mbox}foralln\in \mathbb{N}.$$

Full text

Résumé. On montre un principe de grandes déviations pour une suite de processus ponctuels définit par des mesures de probabilités de Gibbs dans un espace polonais. Il est obtenu comme conséquence d’un principe de Laplace pour des mesures de Gibbs non normalisées. On considère quatre applications: Des mesures de Gibbs conditionnées dans des espaces compacts, des gaz de Coulomb sur des variétés riemanniennes compactes, les mesures de Gibbs habituelles sur l’espace euclidien et les zéros des polynômes aléatoires gaussiens. Finalement, on étudie la généralisation des points Fekete et on prouve une version déterministe du principe de Laplace appelée $\Gamma$-convergence. Notre approche est partiellement inspirée par les travaux de Dupuis et ses coauteurs. C’est notablement naturelle et générale en comparaison avec les stratégies habituelles pour les mesures de Gibbs singulières.

2010 MSC: 60F10; 60K35; 82C22; 30C15

Keywords: Gibbs measure; Coulomb gas; Empirical measure; Large deviation principle; Interacting particle system; Singular potential; Constant curvature; Relative entropy; Random polynomials; Fekete points

1 Introduction

The present article is inspired by part of the work of Dupuis, Laschos and Ramanan on large deviations for a sequence of point processes given by Gibbs measures associated to very general singular two-body interactions [adupuis] but it differs from it in that we take a general sequence of interactions that includes, for instance, the interaction followed by the zeros of random polynomials as in [zelditch]. We follow the philosophy of Dupuis and Ellis [bdupuis] about the use of variational formulas to make plausible and sometimes easier to find a Laplace principle. This philosophy has already been used by Georgii in [zgeorgii] to treat a system of random fields on $\mathbb{Z}^{d}$ with interacting energies that converges uniformly to some limit functional.

We are interested in proving the Laplace principle and the large deviation principle for a very general sequence of energies in a not necessarily compact space. Part of our work has an overlap with the article of Berman [berman2] and it was developed independently. As in [berman2] the interest of this result is the generality of the sequence of energies: they do not need to be made of a two-body interaction potential but they may still be very singular. The key argument of the proof is a well-understood application of Jensen’s inequality together with a general Laplace principle that has as its main ingredient a subadditivity property of the entropy. It is very simple compared to the ad hoc methods used in the usual proofs of the large deviation principles for Coulomb gases such as in [hiaiunitary], [hiaigaussian] [chafai] and [hardy]. In these methods, to prove a large deviation lower bound, the authors usually decompose the space in small regions and this decomposition may not be easy to achieve on a manifold and not so natural to look for. We give a more precise explanation of these methods in Remark LABEL:remark:_usual.

Among the applications we can give we are particularly interested in explaining a simple case inspired by [bermanKEmetrics]. This is the case of a Coulomb gas on a two-dimensional Riemannian manifold. As a second application we study a large deviation principle for a conditional Gibbs measure, i.e. we fix the position of some of the particles and leave the rest of them random. The last applications we discuss are different proofs of already known results such as the special one-dimensional log-gas of [ldpwigner] related to the Gaussian ensembles, the more general one-dimensional log-gas of [guionnet, Section 2.6], the special two-dimensional log-gas [hiaigaussian] related to the Ginibre ensemble of random matrices and its generalization to an $n$-dimensional Coulomb gas in [chafai] and [adupuis], the note in [hardy] about two-dimensional log-gases with a weakly confining potential and the Gaussian random polynomials of [zelditch] and [raphael].

We now explain the contents of each section. The rest of Section 1 will be dedicated to the main definitions and assumptions we will need to state our results. Section LABEL:section:_kbody is about the usual mean-field case, the $k$-body interaction. We give sufficient conditions to be able to apply our result which will become important when we treat the Euclidean space case. In Section LABEL:section:_proof_theorems we begin by giving an idea of the proofs which includes mainly a key variational formula. Then we give the proofs of the main theorem and of its corollary and we finish the section by giving some remarks about the usual proofs we may find in the literature. We discuss four particular examples in Section LABEL:section:_applications. More precisely, the conditional Gibbs measure, the Coulomb gas on a Riemannian manifold, a new way to obtain already known results in the Euclidean space about Coulomb gases and the assertion that the zeros of a Gaussian random polynomials may be treated by our main theorem. We conclude our article with Section LABEL:section:_fekete discussing a deterministic case which falls under the topic of Fekete points and which we consider as the natural deterministic analogue of the Laplace principle.

1.1 Model

Let $M$ be a Polish space, i.e. a separable topological space metrizable by a complete metric. Endow it with the Borel $\sigma$-algebra associated to this topology, i.e. the least $\sigma$-algebra that contains the topology. Denote by $\mathcal{P}(M)$ the space of probability measures in $M$ and endow it with the smallest topology such that $\mu\mapsto\int_{M}fd\mu$ is continuous for every bounded continuous function $f:M\to\mathbb{R}$. With this topology, $\mathcal{P}(M)$ is also a Polish space (see [borkar, Section 2.4]). This is called the weak topology. Suppose we have a sequence $\{W_{n}\}_{n\in\mathbb{N}}$ of symmetric measurable functions

[TABLE]

and a sequence of non-negative numbers $\{\beta_{n}\}_{n\in\mathbb{N}}$ that converges to some $\beta\in(0,\infty]$. Fix a probability measure $\pi\in\mathcal{P}(M)$. We shall be interested in the asymptotic behavior of the Gibbs measures $\gamma_{n}$ defined by

[TABLE]

Define $\tilde{W}_{n}:\mathcal{P}(M)\to(-\infty,\infty]$ by

[TABLE]

Stable sequence (S)****.

We shall say that the sequence $\{W_{n}\}_{n\in\mathbb{N}}$ is a stable sequence if it is uniformly bounded from below, i.e. if there exists $C\in\mathbb{R}$ such that

[TABLE]