Stability of the Mezard-Parisi solution for random manifolds

TL;DR

This paper analyzes the stability of the Parisi solution for random manifolds with replica symmetry breaking, showing non-negativity of Hessian eigenvalues in the continuum limit relevant for low-dimensional manifolds.

Contribution

It extends the stability analysis of the Parisi solution to the case of R-step replica symmetry breaking for random manifolds, especially in the continuum limit.

Findings

Eigenvalues of the Hessian are non-negative in the R→∞ limit.

Stability of the Parisi solution is confirmed for D<2.

General case of R steps of replica symmetry breaking analyzed.

Abstract

The eigenvalues of the Hessian associated with random manifolds are constructed for the general case of $R$ steps of replica symmetry breaking. For the Parisi limit $R\to\infty$ (continuum replica symmetry breaking) which is relevant for the manifold dimension $D<2$, they are shown to be non negative.