WIAS Preprint No. 2198, (2015)

Mean-field interaction of Brownian occupation measures. II: A rigorous construction of the Pekar process



Authors

  • Bolthausen, Erwin
  • König, Wolfgang
  • Mukherjee, Chiranjib

2010 Mathematics Subject Classification

  • 60J65 60J55 60F10

Keywords

  • Polaron problem, Gibbs measures, large deviations, Coulomb functional, tightness

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 [DV83] in terms of the Pekar variational formula, which coincides with the behavior of the partition function corresponding to the polaron problem under strong coupling. Based on this, Spohn ([Sp87]) 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 itPekar 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 [MV14], its extension to the uniform strong metric for the singular Coulomb interaction carried out in [KM15], as well as an idea inspired by a itpartial path exchange argument appearing in [BS97]

Appeared in

  • Comm. Pure Appl. Math., 70 (2017), pp. 1598--1629.

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