Glassy random matrix models.
This paper discusses random matrix models that exhibit the unusual phenomena of having multiple solutions at the same point in phase space. These matrix models have gaps in their spectrum or density of eigenvalues. The free energy and certain correlation functions of these models show differences for the different solutions. This study presents evidence for the presence of multiple solutions both analytically and numerically. As an example this paper discusses the double-well matrix model with potential V(M)=-(mu/2)M(2)+(g/4)M(4), where M is a random N x N matrix (the M(4) matrix model) as well as the Gaussian Penner model with V(M)=(mu/2)M(2)-t ln M. First this paper studies what these multiple solutions are in the large N limit using the recurrence coefficient of the orthogonal polynomials. Second it discusses these solutions at the nonperturbative level to bring out some differences between the multiple solutions. Also presented are the two-point density-density correlation functions, which further characterize these models in a different universality class. A motivation for this work is that variants of these models have been conjectured to be models of certain structural glasses in the high temperature phase.