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

TL;DR

This paper proves that the effective mass of the Fröhlich Polaron diverges as a quartic function of the coupling constant, confirming a long-standing theoretical prediction and introducing new analytical techniques.

Contribution

It establishes the sharp quartic divergence rate of the Polaron effective mass and introduces a novel method based on Gaussian and Poisson point process representations.

Findings

Effective mass diverges as er ourth power of coupling or large er.

Method demonstrates natural emergence of quartic divergence rate.

Results include explicit identification of local interval process and monotonicity of effective mass.

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.