Critical Behaviour of the Number of Minima of a Random Landscape at the Glass Transition Point and the Tracy-Widom distribution

TL;DR

This paper investigates the critical behavior of the number of minima in a random Gaussian landscape at a glass transition, revealing a Tracy-Widom distribution governs the fluctuations near the critical point.

Contribution

It establishes a connection between minima count in a random landscape and Tracy-Widom distribution, providing detailed analysis of the transition behavior in high-dimensional systems.

Findings

Number of minima drops from exponential to near one at transition

Width of critical region scales as N^{-1/3}

Minima distribution converges to Tracy-Widom form near criticality

Abstract

We exploit a relation between the mean number $N_{m}$ of minima of random Gaussian surfaces and extreme eigenvalues of random matrices to understand the critical behaviour of $N_{m}$ in the simplest glass-like transition occuring in a toy model of a single particle in $N$-dimensional random environment, with $N\gg 1$. Varying the control parameter $\mu$ through the critical value $\mu_c$ we analyse in detail how $N_{m}(\mu)$ drops from being exponentially large in the glassy phase to $N_{m}(\mu)\sim 1$ on the other side of the transition. We also extract a subleading behaviour of $N_{m}(\mu)$ in both glassy and simple phases. The width $\delta{\mu}/\mu_c$ of the critical region is found to scale as $N^{-1/3}$ and inside that region $N_{m}(\mu)$ converges to a limiting shape expressed in terms of the Tracy-Widom distribution.