We consider the Fr\"ohlich model of the Polaron whose path integral formulation leads to the transformed path measure $$ \widehat{ \mathbb P}_{ \alpha,T}(\mathrm d\omega)= Z_{ \alpha,T}^{ -1}\,\, \exp\bigg\{ \frac{ \alpha}{ 2}\int_{ -T}^T\int_{ -T}^T\frac{ e^{ -|t-s|}}{ |\omega(t)-\omega(s)|} \, d s \, d t\bigg\}\,\mathbb P(\mathrm d\omega) $$ 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_{ \alpha,T}$ is the partition function or the normalizing constant and $\alpha>0$ is a constant. The Polaron measure reflects a self attractive interaction. According to a conjecture of Pekar that was proved in [DV83] $$ g_0=\lim_{ \alpha \to\infty}\frac{ 1}{ \alpha^2}\bigg[\lim_{ T\to\infty}\frac{ \log Z_{ \alpha,T}}{ 2T}\bigg] $$ exists and has a variational formula. In this article we show that for any $\alpha>0$, the infinite-volume limit $\widehat{ \mathbb P}_{ \alpha}=\lim_{ T\to\infty}\widehat{ \mathbb P}_{ \alpha,T}$ exists which is also identified explicitly. As a corollary, we deduce the central limit theorem (for any $\alpha>0$ and as $T\to\infty$) for the distribution of $\frac{ \omega(T)-\omega(-T)}{ \sqrt{ 2T}}$ both under the finite-volume Polaron measure $\widehat{ \mathbb P}_{ \alpha,T}$ and its infinite-volume counterpart $\widehat{ \mathbb P}_\alpha$, and obtain an expression for the limiting variance.
Categories: math.PR math-ph math.MP