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

Wednesday, 18 November, 2015

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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 \cite{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 (\cite{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 {\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 \cite{MV14}, its extension to the uniform strong metric for the singular Coulomb interaction carried out in \cite{KM15}, as well as an idea inspired by a {\it{partial path exchange}} argument appearing in \cite{BS97}.


Erwin Bolthausen
Wolfgang Koenig
Chiranjib Mukherjee