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The Larkin Mass and Replica Symmetry Breaking in the Elastic Manifold

Gerard Ben Arous, Pax Kivimae

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].

The Larkin Mass and Replica Symmetry Breaking in the Elastic Manifold

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].

Paper Structure

This paper contains 11 sections, 43 theorems, 292 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results for the Elastic Manifold Model
  3. Results on the Replica Symmetric phase
  4. Results for the Spherical Model
  5. Overview
  6. History and Related Works
  7. Proof of Theorems \ref{['theorem:intro:main:sphere']}, \ref{['theorem:intro:parisi-measure:sphere']}, and \ref{['theorem:intro:main:sphere-generic']}
  8. Minimization Identities for the Parisi Functional
  9. The Euclidean Model and Concavity
  10. Proof of Theorem \ref{['theorem:intro:main:Euclidean parameters identification']}
  11. The RS Regime and the Larkin Mass

Key Result

Theorem 1.1

We have a.s. that where $F_{\Omega}(\beta,t,\mu)$ is an explicit deterministic quantity.

Figures (1)

  • Figure 1: The phase diagram for $L^d=1$ and $B(x)=e^{-x}+e^{-8x}$ in terms of $(\beta,\mu)$. Solutions to (\ref{['eqn:intro:fake-larkin']}) are plotted in orange, the AT-line is in green, and the fixed inverse-temperature $\beta_{DW}$ is in red.

Theorems & Definitions (75)

  • Theorem 1.1: Paper1, Theorem I.1.3
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.4: A New Parisi Formula
  • Theorem 1.5: A Variational Form of The Parisi Formula
  • Theorem 1.6: Identification of the Overlap Distribution and Effective Radius
  • Theorem 1.7: Characterization of The RS-Phase
  • Corollary 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 65 more