High-Dimensional Random Fields and Random Matrix Theory

TL;DR

This paper combines high-dimensional Kac-Rice formulae with Random Matrix Theory to analyze the topology of random energy landscapes in high-dimensional Gaussian fields, revealing universal features near glass transitions.

Contribution

It introduces a novel analytical framework linking Kac-Rice and RMT to study stationary points in high-dimensional Gaussian fields and elucidates the topology trivialization near glass transitions.

Findings

Identification of the role of GOE edge scaling and Tracy-Widom distribution in topology trivialization.

Quantitative description of the reduction in stationary points near the glass transition.

Universal features of landscape topology captured through RMT techniques.

Abstract

Our goal is to discuss in detail the calculation of the mean number of stationary points and minima for random isotropic Gaussian fields on a sphere as well as for stationary Gaussian random fields in a background parabolic confinement. After developing the general formalism based on the high-dimensional Kac-Rice formulae we combine it with the Random Matrix Theory (RMT) techniques to perform analysis of the random energy landscape of $p-$spin spherical spinglasses and a related glass model, both displaying a zero-temperature one-step replica symmetry breaking glass transition as a function of control parameters (e.g. a magnetic field or curvature of the confining potential). A particular emphasis of the presented analysis is on understanding in detail the picture of "topology trivialization" (in the sense of drastic reduction of the number of stationary points) of the landscape which takes place in the vicinity of the zero-temperature glass transition in both models. We will reveal the important role of the GOE "edge scaling" spectral region and the Tracy-Widom distribution of the maximal eigenvalue of GOE matrices for providing an accurate quantitative description of the universal features of the topology trivialization scenario.