Large deviation principle for empirical fields of Log and Riesz gases

By Thomas Leblé, Sylvia Serfaty. Inventiones Mathematicae, 2015

173 citations recorded by the source corpus.

We study a system of N particles with logarithmic, Coulomb or Riesz pairwise interactions, confined by an external potential. We examine a microscopic quantity, the tagged empirical field, for which we prove a large deviation principle at speed N. The rate function is the sum of an entropy term, the specific relative entropy, and an energy term, the renormalized energy introduced in previous works, coupled by the temperature. We deduce a variational property of the sine-beta processes which arise in random matrix theory. We also give a next-to-leading order expansion of the free energy of the system, proving the existence of the thermodynamic limit.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/04867ed8df802b18d0e3caa4f9d59670a412fc06

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Mathematical Physics

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

Trainability and Accuracy of Artificial Neural Networks: An Interacting Particle System Approach (2018, methodology): “Using the interchangeability of the n interacting particles/parameters in the neural representation, we focus on their empirical distribution and analyze its properties when n is large using standard limit theorems [22, 23, 31, 32].”
Neural Networks as Interacting Particle Systems: Asymptotic Convexity of the Loss Landscape and Universal Scaling of the Approximation Error (2018, methodology): “We also consider the empirical distribution of the n interacting particles / parameters and analyze the properties of this distribution when the number n is large using standard limit theorems [17, 18, 26, 27].”
The Crystallization Conjecture: A Review (2015, background): “The link between a trapped Coulomb gas in the mean-field limit and the positive-temperature Jellium problem was recently studied in [150, 149], similarly to what we discussed in (43).”
Quantitative estimates of propagation of chaos for stochastic systems with W-1,∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d (2017, methodology): “See also some applications in the context of Log and Riesz Gases in [64].”