SwissMAP Logo
Log in
  • About us
    • Organization
    • Professors
    • Senior Researchers
    • Postdocs
    • PhD Students
    • Alumni
  • News & Events
    • News
    • Events
    • Online Events
    • Videos
    • Newsletters
    • Press Coverage
    • Perspectives Journal
    • Interviews
  • Research
    • Basic Notions
    • Field Theory
    • Geometry, Topology and Physics
    • Quantum Systems
    • Statistical Mechanics
    • String Theory
    • Publications
    • SwissMAP Research Station
  • Awards, Visitors & Vacancies
    • Awards
    • Innovator Prize
    • Visitors
    • Vacancies
  • Outreach & Education
    • Masterclasses & Doctoral Schools
    • Mathscope
    • Maths Club
    • Athena Project
    • ETH Math Youth Academy
    • SPRING
    • Junior Euler Society
    • General Relativity for High School Students
    • Outreach Resources
    • Exhibitions
    • Previous Programs
    • Events in Outreach
    • News in Outreach
  • Equal Opportunities
    • Mentoring Program
    • Financial Support
    • SwissMAP Scholars
    • Events in Equal Opportunities
    • News in Equal Opportunities
  • Contact
    • Corporate Design
  • Basic Notions
  • Field Theory
  • Geometry, Topology and Physics
  • Quantum Systems
  • Statistical Mechanics
  • String Theory
  • Publications
  • SwissMAP Research Station

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

Erwin Bolthausen, Wolfgang Koenig, Chiranjib Mukherjee

18/11/15 Published in : arXiv:1511.05921

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}.

Entire article

Research project(s)

  • Statistical Mechanics

A note on Ising random currents, Ising-FK, loop-soups and the Gaussian free field

Props of ribbon graphs, involutive Lie bialgebras and moduli spaces of curves

  • Leading house

  • Co-leading house


The National Centres of Competence in Research (NCCRs) are a funding scheme of the Swiss National Science Foundation

© SwissMAP 2023 - All rights reserved