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