Compact support probability distributions
in random matrix theory.
Graziano Vernizzi
Abstract
A random matrix statistical ensemble is defined by the joint probability
density for the independent entries of the matrix. In this framework a very
popular probability density is the "canonical" one $\sim \exp (-n TrV(M))$,
where V(M) is a polynomial and M a $n \times n$ matrix. We study two
"generalized restricted trace ensembles" defined by the probability densities
$\sim \delta (A^2-n TrV(M))$ and $\sim \theta (A^2-n TrV(M))$: they are a
generalization of matrix ensembles studied long ago by Rosenzweig and Bronk,
where only the case $V(x)=x^2$ was considered. Restricted trace ensembles are
interesting for several features: the interaction among eigenvalues is
introduced through a constraint very similiar to the non linear sigma model
in quantum field theory, the spectral density has compact support both for
finite n and in the "large-n" limit, and they relate to "canonical" probability
density just in the same way as the microcanonical ensemble is related to the
canonical ensemble in statistical mechanics. Nevertheless, this relation is
not trivial and non linear as it appears by studying the phase diagram of
these models.
Elenco dei partecipanti al convegno di Bari.