A method for constructing random matrix models of disordered bosons

TL;DR

This paper introduces a new method for constructing random matrix models of disordered bosons, focusing on eigenvalue properties and geometric structures, with explicit results in a simplified case.

Contribution

It proposes a novel approach to build G-invariant ensembles of matrices with purely imaginary eigenvalues using moment map images from complex geometric phase spaces.

Findings

Explicit spectral measure description for the n=1 case

Construction of G-invariant matrix ensembles with stability conditions

Connection between matrix models and complex geometry

Abstract

Random matrix models of disordered bosons consist of matrices in the Lie algebra g=sp_n(R). Assuming dynamical stability, their eigenvalues are required to be purely imaginary. Here a method is proposed for constructing ensembles (E,P) of G-invariant sets E of such matrices with probability measures P. These arise as moment map direct images from phase spaces X which play an important role in complex geometry and representation theory. In the toy-model case of n=1, where X is the complex bidisk and P is the direct image of the uniform measure, an explicit description of the spectral measure is given.