Identification of the Polaron measure I: Fixed coupling regime and the central limit theorem for large times

Abstract

We consider the Fröhlich model of the Polaron whose path integral formulation leads to the transformed path measure $$ \widehat{\mathbb P}_{α,T}(\mathrm dω)= Z_{α,T}^{-1}\,\, \exp\bigg\{\fracα{2}\int_{-T}^T\int_{-T}^T\frac{e^{-|t-s|}}{|ω(t)-ω(s)|} \, d s \, d t\bigg\}\,\mathbb P(\mathrm dω) $$ with respect to $\mathbb P$ which governs the law of the increments of the three dimensional Brownian motion on a finite interval $[-T,T]$, and $ Z_{α,T}$ is the partition function or the normalizing constant and $α>0$ is a constant. The Polaron measure reflects a self attractive interaction. According to a conjecture of Pekar that was proved in [DV83] $$ g_0=\lim_{α\to\infty}\frac{1}{α^2}\bigg[\lim_{T\to\infty}\frac{\log Z_{α,T}}{2T}\bigg] $$ exists and has a variational formula. In this article we show that for any $α>0$, the infinite-volume limit $\widehat{\mathbb P}_α=\lim_{T\to\infty}\widehat{\mathbb P}_{α,T}$ exists which is also identified explicitly. As a corollary, we deduce the central limit theorem (for any $α>0$ and as $T\to\infty$) for the distribution of $\frac{ω(T)-ω(-T)}{\sqrt{2T}}$ both under the finite-volume Polaron measure $\widehat{\mathbb P}_{α,T}$ and its infinite-volume counterpart $\widehat{\mathbb P}_α$, and obtain an expression for the limiting variance.