14 research outputs found
Two-band random matrices
Spectral correlations in unitary invariant, non-Gaussian ensembles of large
random matrices possessing an eigenvalue gap are studied within the framework
of the orthogonal polynomial technique. Both local and global characteristics
of spectra are directly reconstructed from the recurrence equation for
orthogonal polynomials associated with a given random matrix ensemble. It is
established that an eigenvalue gap does not affect the local eigenvalue
correlations which follow the universal sine and the universal multicritical
laws in the bulk and soft-edge scaling limits, respectively. By contrast,
global smoothed eigenvalue correlations do reflect the presence of a gap, and
are shown to satisfy a new universal law exhibiting a sharp dependence on the
odd/even dimension of random matrices whose spectra are bounded. In the case of
unbounded spectrum, the corresponding universal `density-density' correlator is
conjectured to be generic for chaotic systems with a forbidden gap and broken
time reversal symmetry.Comment: 12 pages (latex), references added, discussion enlarge
Glassy Random Matrix Models
This paper discusses Random Matrix Models which 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. Here I present evidence for the presence of multiple
solutions both analytically and numerically.
As an example I discuss the double well matrix model with potential where is a random matrix (the
matrix model) as well as the Gaussian Penner model with . First I study what these multiple solutions are in the large
limit using the recurrence coefficient of the orthogonal polynomials.
Second I discuss these solutions at the non-perturbative level to bring out
some differences between the multiple solutions. I also present the two-point
density-density correlation functions which further characterizes these models
in a new university 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.Comment: 25 pages, Latex, 7 Figures, to appear in PR