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

Abstract

AbstractWe consider the Fröhlich model of the polaron, whose path integral formulation leads to the transformed path measureurn:x-wiley:15347710:media:cpa21858:cpa21858-math-0001with 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], urn:x-wiley:15347710:media:cpa21858:cpa21858-math-0002 exists and has a variational formula. In this article we show that when α > 0 is either sufficiently small or sufficiently large, the limit exists, which is also identified explicitly. As a corollary, we deduce the central limit theorem for urn:x-wiley:15347710:media:cpa21858:cpa21858-math-0003 under and obtain an expression for the limiting variance. © 2019 the Authors. Communications on Pure and Applied Mathematics is published by the Courant Institute of Mathematical Sciences and Wiley Periodicals, LLC.