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

Chiranjib Mukherjee, Srinivasa 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)
JournalCommunications on Pure and Applied Mathematics
DOIs
StateAccepted/In press - Jan 1 2019

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Polaron
Brownian movement
Central limit theorem
Normalizing Constant
Curvilinear integral
Partition Function
Increment
Brownian motion
Deduce
Corollary
Limiting
Three-dimensional
Path
Interval
Formulation
Interaction

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Identification of the Polaron Measure I : Fixed Coupling Regime and the Central Limit Theorem for Large Times. / Mukherjee, Chiranjib; Varadhan, Srinivasa.

In: Communications on Pure and Applied Mathematics, 01.01.2019.

Research output: Contribution to journalArticle

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