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

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 Z_{α, 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.