Mean-Field interacton of Brownian occupation measures. II: A rigorous construction of the Pekar process

TL;DR

This paper rigorously constructs the Pekar process as the limit of mean-field measures for interacting Brownian paths with Coulomb potential, advancing the understanding of the polaron problem's path measure approximation.

Contribution

It provides a rigorous construction of the Pekar process by analyzing the convergence of occupation measures in a mean-field Brownian interaction model.

Findings

Convergence of normalized occupation measures to a mixture of Pekar variational maximizers.

Rigorous construction of the Pekar process from mean-field path measures.

Extension of large deviation techniques to singular Coulomb interactions.

Abstract

We consider mean-field interactions corresponding to Gibbs measures on interacting Brownian paths in three dimensions. The interaction is self-attractive and is given by a singular Coulomb potential. The logarithmic asymptotics of the partition function for this model were identified in the 1980s by Donsker and Varadhan [6] in terms of the {\it{Pekar variational formula}}, which coincides with the behavior of the partition function of the {\it{polaron problem}} under strong coupling. Based on this, in 1986 Spohn [14] made a heuristic observation that the strong coupling behavior of the polaron path measure, on certain time scales, should resemble a process, named as the {\it{Pekar process}}, whose distribution could somehow be guessed from the limiting asymptotic behavior of the mean-field measures under interest, whose rigorous analysis remained open. The present paper is devoted to a precise analysis of these mean-field path measures and convergence of the normalized occupation measures towards an explicit mixture of the maximizers of the Pekar variational problem. This leads to a rigorous construction of the aforementioned Pekar process and hence, is a contribution to the understanding of the "mean-field approximation" of the polaron problem on the level of path measures. The method of our proof is based on the compact large deviation theory developed in [11], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [8], as well as an idea inspired by a {\it{partial path exchange}} argument appearing in [1].