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

We exploit a relation between the mean number Nₘ of minima of random Gaussian surfaces and extreme eigenvalues of random matrices to understand the critical behavior of Nₘ in the simplest glasslike transition occuring in a toy model of a single particle in an N-dimensional random environment, with N1. Varying the control parameter μ through the critical value μc we analyze in detail how Nₘ(μ) drops from being exponentially large in the glassy phase to Nₘ(μ)~1 on the other side of the transition. We also extract a subleading behavior of Nₘ(μ) in both glassy and simple phases. The width δμ/μc of the critical region is found to scale as N^-1/3 and inside that region Nₘ(μ) converges to a limiting shape expressed in terms of the Tracy-Widom distribution.