Almost Quartic Lower Bound for the Fr\"{o}hlich Polaron's Effective Mass via Gaussian Domination

TL;DR

This paper establishes a nearly quartic lower bound on the effective mass of the Fröhlich polaron at large coupling, matching longstanding theoretical predictions and complementing recent upper bounds, using Gaussian correlation inequalities.

Contribution

It provides the first nearly quartic lower bound for the polaron's effective mass, advancing understanding of its behavior at strong coupling.

Findings

Effective mass grows at least as rac{ ext{constant} imes ext{alpha}^4}{( ext{log alpha})^6} for large alpha

Matches the predicted quartic growth rate by Landau and Pekar from 1948

Complements recent sharp upper bounds by Brooks and Seiringer

Abstract

We prove the Fr\"{o}hlich polaron has effective mass at least $\frac{\alpha^4}{(\log \alpha)^6}$ when the coupling strength $\alpha$ is large. This nearly matches the quartic growth rate $C_*\alpha^4$ predicted by Landau and Pekar in 1948 and complements a recent sharp upper bound of Brooks and Seiringer. Our proof works with the path integral formulation of the problem and systematically applies the Gaussian correlation inequality to exploit quasi-concavity of the interaction terms.