Distribution of rare saddles in the $p$-spin energy landscape

TL;DR

This paper analyzes the distribution and rarity of index-1 saddles near local minima in the $p$-spin energy landscape, providing insights into the landscape's structure and rare configurations through large deviation calculations.

Contribution

It introduces a method to compute the distribution of saddles around minima and characterizes rare saddle configurations via large deviation probabilities.

Findings

Distribution of saddles as a function of distance and energy

Identification of rare saddles with subdominant number

Large deviation probabilities for eigenvalues and eigenvectors of perturbed GOE matrices

Abstract

We compute the statistical distribution of index-1 saddles surrounding a given local minimum of the $p$-spin energy landscape, as a function of their distance to the minimum in configuration space and of the energy of the latter. We identify the saddles also in the region of configuration space in which they are subdominant in number (i.e., rare) with respect to local minima, by computing large deviation probabilities of the extremal eigenvalues of their Hessian. As an independent result, we determine the joint large deviation probability of the smallest eigenvalue and eigenvector of a GOE matrix perturbed with both an additive and multiplicative finite-rank perturbation.