Critical Behaviour 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 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 $\delta \mu \mathrm{/}{\mu }_c$ of the critical region is found to scale as $N^{-1\mathrm{/}3}$ and inside that region $N_m(\mu )$ converges to a limiting shape expressed in terms of the Tracy-Widom distribution.