High-Dimensional Random Fields and Random Matrix Theory∗

Our goal is to discuss in detail the calculation of the mean num-ber of stationary points and minima for random isotropic Gaussian fields on a sphere as well as for stationary Gaussian random fields in a background parabolic confinement. After developing the general for-malism based on the high-dimensional Kac-Rice formulae we combine it with the Random Matrix Theory (RMT) techniques to perform analysis of the random energy landscape of p−spin spherical spin-glasses and a related glass model, both displaying a zero-temperature one-step replica symmetry breaking glass transition as a function of control parameters (e.g. a magnetic field or curvature of the confining potential). A particular emphasis of the presented analysis is on un-derstanding in detail the picture of ”topology trivialization ” (in the sense of drastic reduction of the number of stationary points) of the landscape which takes place in the vicinity of the zero-temperature glass transition in both models. We will reveal the important role of the GOE ”edge scaling ” spectral region and the Tracy-Widom dis-tribution of the maximal eigenvalue of GOE matrices for providing an accurate quantitative description of the universal features of the topology trivialization scenario. ∗Dedicated to Prof. Leonid Pastur on the occassion of his 75th birthday 1 a