Bootstrapping the critical behavior of multi-matrix models

TL;DR

This paper applies a positivity-based bootstrap method to estimate critical points and exponents in multi-matrix models, revealing potential new continuum limits and providing explicit free energy expansions.

Contribution

It extends the bootstrap with positivity technique to complex multi-matrix models, including conjectures on new universality classes and explicit free energy calculations.

Findings

Confirmed critical phenomena in quartic single matrix model

Estimated string susceptibility exponents for 2-matrix models

Derived explicit free energy series expansions

Abstract

Given a matrix model, by combining the Schwinger-Dyson equations with positivity constraints on its solutions, in the large $N$ limit one is able to obtain explicit and numerical bounds on its moments. This technique is known as bootstrapping with positivity. In this paper we use this technique to estimate the critical points and exponents of several multi-matrix models. As a proof of concept, we first show it can be used to find the well-studied quartic single matrix model's critical phenomena. We then apply the method to several similar ``unsolved" 2-matrix models with various quartic interactions. We conjecture and present strong evidence for the string susceptibility exponent for some of these models to be $\gamma = 1/2$, which heuristically indicates that the continuum limit will likely be the Continuum Random Tree. For the other 2-matrix models, we find estimates of new string susceptibility exponents that may indicate a new continuum limit. We then study an unsolved 3-matrix model that generalizes the 3-colour model with cubic interactions. Additionally, for all of these models, we are able to derive explicitly the first several terms of the free energy in the large $N$ limit as a power series expansion in the coupling constants at zero by exploiting the structure of the Schwinger-Dyson equations.