The Larkin Mass and Replica Symmetry Breaking in the Elastic Manifold

TL;DR

This paper analyzes the free energy of the Elastic Manifold model, providing a simplified variational formula, characterizing replica symmetry breaking, and confirming the significance of the Larkin mass as a critical threshold.

Contribution

It introduces a simplified variational approach and characterizes the replica symmetry breaking phase in the Elastic Manifold model, linking it to the Larkin mass.

Findings

Complete characterization of the replica symmetry breaking phase.

Confirmation of the Larkin mass as a critical boundary.

Interpretation of model statistics in terms of overlap and effective radius.

Abstract

This is the second of a series of three papers about the Elastic Manifold model. This classical model proposes a rich picture due to the competition between the inherent disorder and the smoothing effect of elasticity. In this paper, we analyze our variational formula for the free energy obtained in our first companion paper [16]. We show that this variational formula may be simplified to one which is solved by a unique saddle point. We show that this saddle point may be solved for in terms of the corresponding critical point equation. Moreover, its terms may be interpreted in terms of natural statistics of the model: namely the overlap distribution and effective radius of the model at a given site. Using this characterization, obtain a complete characterization of the replica symmetry breaking phase. From this we are able to confirm a number of physical predictions about this boundary, namely those involving the Larkin mass [6, 53, 54], an important critical mass for the system. The zero-temperature Larkin mass has recently been shown to be the topological trivialization threshold, following work of Fyodorov and Le Doussal [37, 38], made rigorous by the first author, Bourgade and McKenna [12, 13].