Variational Master Field for Large-N Interacting Matrix Models - Free Random Variables on Trial

TL;DR

This paper introduces a novel variational method for large-N matrix models using free random variables, demonstrating strong agreement with exact results across various models.

Contribution

It develops a new variational approach leveraging free random matrices as the variational space for large-N matrix models.

Findings

High accuracy of variational solutions compared to Monte Carlo results

Applicable to classical and quantum matrix models with different interactions

Provides a new analytical tool for studying large-N matrix dynamics

Abstract

Matrices are said to behave as free non-commuting random variables if the action which governs their dynamics constrains only their eigenvalues, i.e. depends on traces of powers of individual matrices. The authors use recently developed mathematical techniques in combination with a standard variational principle to formulate a new variational approach for matrix models. Approximate variational solutions of interacting large-N matrix models are found using the free random matrices as the variational space. Several classes of classical and quantum mechanical matrix models with different types of interactions are considered and the variational solutions compared with exact Monte Carlo and analytical results. Impressive agreement is found in a majority of cases.