[Paper Review] Large deviations for empirical measures generated by Gibbs measures with singular energy functionals

Paul Dupuis, Vaios Laschos+1 more|arXiv (Cornell University)|Jan 1, 2015

Markov Chains and Monte Carlo Methods8 citations

TL;DR

This paper establishes large deviation principles (LDPs) for empirical measures of n-particle systems governed by Gibbs distributions with singular energy functionals, using weak convergence methods to derive LDPs under Wasserstein-type topologies. It provides a unified framework for all speeds, including the case bn = n, where an entropic term appears in the rate function, resolving an open question in prior work on confined particles with singular repulsion.

ABSTRACT

We establish large deviation principles (LDPs) for empirical measures associated with a sequence of Gibbs distributions on n-particle configurations, each of which is defined in terms of an inverse temperature bn and an energy functional that is the sum of a (possibly singular) interaction and confining potential. Under fairly general assumptions on the potentials, we establish LDPs both with speeds (bn)/(n) { extregistered} { extyen}, in which case the rate function is expressed in terms of a functional involving the potentials, and with the speed bn =n, when the rate function contains an additional entropic term. Such LDPs are motivated by questions arising in random matrix theory, sampling and simulated annealing. Our approach, which uses the weak convergence methods developed in ``A weak convergence approach to the theory of large deviations", establishes large deviation principles with respect to stronger, Wasserstein-type topologies, thus resolving an open question in "First-order global asymptotics for confined particles with singular pair repulsion". It also provides a common framework for the analysis of LDPs with all speeds, and includes cases not covered due to technical reasons in previous works.

Motivation & Objective

• To establish large deviation principles (LDPs) for empirical measures of n-particle systems with singular interaction and confining potentials under general conditions.
• To resolve an open question regarding LDPs in Wasserstein-type topologies for confined particles with singular pair repulsion.
• To provide a unified framework for analyzing LDPs across all speeds, including the critical case bn = n.
• To incorporate entropic terms in the rate function when the speed is bn = n, extending previous results.
• To apply the results to problems in random matrix theory, sampling, and simulated annealing, motivated by physical and statistical mechanics contexts.

Proposed method

• Uses weak convergence methods from the theory of large deviations to derive LDPs with respect to stronger, Wasserstein-type topologies.
• Analyzes Gibbs measures defined by inverse temperature bn and energy functionals combining singular interactions and confining potentials.
• Derives the rate function in two regimes: with speed (bn)/n, where the rate function depends on the potentials, and with speed bn = n, where an additional entropic term appears.
• Applies a variational approach to characterize the rate function in terms of free energy functionals involving the potentials and entropy.
• Establishes convergence under general assumptions on the potentials, including singularities, without requiring technical restrictions that limited prior works.
• Uses the framework to unify and extend previous results, particularly those in the context of first-order global asymptotics for confined particles.

Experimental results

Research questions

• RQ1 What large deviation principles govern the empirical measures of n-particle systems with singular energy functionals under general potential assumptions?
• RQ2 How does the rate function change when the speed is (bn)/n versus bn = n, particularly regarding the appearance of entropic terms?
• RQ3 Can LDPs be established under stronger topologies such as Wasserstein-type metrics for singular Gibbs systems?
• RQ4 What is the role of the confining potential and singular interaction in shaping the large deviation behavior of empirical measures?
• RQ5 How can the weak convergence approach be leveraged to unify and extend existing LDP results for particle systems with singular interactions?

Key findings

• The paper establishes large deviation principles for empirical measures with respect to Wasserstein-type topologies, resolving an open question in the analysis of confined particles with singular repulsion.
• For the speed (bn)/n, the rate function is expressed as a functional involving the interaction and confining potentials, without an entropic term.
• When the speed is bn = n, the rate function includes an additional entropic term, reflecting the system's statistical fluctuations.
• The framework applies to a broad class of potentials, including singular interactions, without the technical restrictions that limited earlier studies.
• The results provide a common theoretical foundation for analyzing LDPs across all speeds, unifying previously fragmented approaches.
• The approach extends to applications in random matrix theory, sampling, and simulated annealing, offering a rigorous large deviation description for these settings.