HighDimensional Random Fields and Random Matrix Theory
Abstract
Our goal is to discuss in detail the calculation of the mean number 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 formalism based on the highdimensional KacRice formulae we combine it with the Random Matrix Theory (RMT) techniques to perform analysis of the random energy landscape of $p$spin spherical spinglasses and a related glass model, both displaying a zerotemperature onestep 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 understanding 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 zerotemperature glass transition in both models. We will reveal the important role of the GOE "edge scaling" spectral region and the TracyWidom distribution of the maximal eigenvalue of GOE matrices for providing an accurate quantitative description of the universal features of the topology trivialization scenario.
 Publication:

arXiv eprints
 Pub Date:
 July 2013
 DOI:
 10.48550/arXiv.1307.2379
 arXiv:
 arXiv:1307.2379
 Bibcode:
 2013arXiv1307.2379F
 Keywords:

 Mathematical Physics;
 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics;
 Mathematics  Probability
 EPrint:
 40 pages