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

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Abstract
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 N$\gg${}1. Varying the control parameter $\mu${} through the critical value $\mu${}c we analyze in detail how Nm($\mu${}) drops from being exponentially large in the glassy phase to Nm($\mu${})$\sim${}1 on the other side of the transition. We also extract a subleading behavior of Nm($\mu${}) 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 Nm($\mu${}) converges to a limiting shape expressed in terms of the Tracy-Widom distribution.

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