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We exploit a relation between the mean number $N_{m}$ of minima of randomGaussian surfaces and extreme eigenvalues of random matrices to understand thecritical behaviour of $N_{m}$ in the simplest glass-like transition occuring ina 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 exponentiallylarge in the glassy phase to $N_{m}(\mu)\sim 1$ on the other side of thetransition. We also extract a subleading behaviour of $N_{m}(\mu)$ in bothglassy and simple phases. The width $\delta{\mu}/\mu_c$ of the critical regionis found to scale as $N^{-1/3}$ and inside that region $N_{m}(\mu)$ convergesto a limiting shape expressed in terms of the Tracy-Widom distribution.

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