Identification of the Polaron Measure I: Fixed Coupling Regime and the Central Limit Theorem for Large Times

Chiranjib Mukherjee, S. R.S. Varadhan

Research output: Contribution to journalArticle

Abstract

We consider the Fröhlich model of the polaron, whose path integral formulation leads to the transformed path measure (Formula presented.) with respect to ℙ that governs the law of the increments of the three-dimensional Brownian motion on a finite interval [−T, T], and Zα, T is the partition function or the normalizing constant and α > 0 is a constant, or the coupling parameter. The polaron measure reflects a self-attractive interaction. According to a conjecture of Pekar that was proved in [9], (Formula presented.) exists and has a variational formula. In this article we show that when α > 0 is either sufficiently small or sufficiently large, the limit (Formula presented.) exists, which is also identified explicitly. As a corollary, we deduce the central limit theorem for (Formula presented.) under (Formula presented.) and obtain an expression for the limiting variance.

Original languageEnglish (US)
Pages (from-to)350-383
Number of pages34
JournalCommunications on Pure and Applied Mathematics
Volume73
Issue number2
DOIs
StatePublished - Feb 1 2020

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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