Counting Multiple Solutions in Glassy Random Matrix Models

TL;DR

This paper introduces a method to count multiple solutions in glassy random matrix models by relating it to a moment problem, revealing exponential growth in solutions and emphasizing the role of asymmetry.

Contribution

It reduces the problem of counting solutions to a moment problem and demonstrates the exponential growth of solutions with matrix size, highlighting the importance of asymmetry.

Findings

Number of solutions grows exponentially with matrix size N.

Asymmetry (tapping) is crucial for finding multiple solutions.

Implications for supercooled liquids in glassy models.

Abstract

This is a first step in counting the number of multiple solutions in certain glassy random matrix models introduced in refs. \cite{d02}. We are able to do this by reducing the problem of counting the multiple solutions to that of a moment problem. More precisely we count the number of different moments when we introduce an asymmetry (tapping) in the random matrix model and then take it to vanish. It is shown here that the number of moments grows exponentially with respect to N the size of the matrix. As these models map onto models of structural glasses in the high temperature phase (liquid) this may have interesting implications for the supercooled liquid phase in these spin glass models. Further it is shown that the nature of the asymmetry (tapping) is crutial in finding the multiple solutions. This also clarifies some of the puzzles we raised in ref. \cite{bd99}.