Effective Mass of the Fr\"ohlich Polaron and the Landau-Pekar-Spohn Conjecture

Abstract

We prove that there is a constant $\overline C\in (0,\infty)$ such that the effective mass $m(\alpha)$ of the Fr\"ohlich Polaron satisfies $m(\alpha) \geq \overline C \alpha^4$, which is sharp according to a long-standing prediction of Landau-Pekar [19] from 1948 and of Spohn [36] from 1987. The method of proof, which demonstrates how the sharp quartic divergence rate of $m(\alpha)$ appears in a natural way, is based on analyzing the Gaussian representation of the Polaron measure and that of the associated tilted Poisson point process developed in [26]. Additionally, our technique here leads to accompanying results including, 1) an explicit identification of local interval process from [26] in the strong coupling limit in terms of functionals of the Pekar process [27], 2) strict monotonicity of the effective mass $m(\alpha)$ for all $\alpha>0$ and 3) the quartic divergence of $m(\alpha)$ for a generalized class of Polaron type interactions.