Large N Optimization for multi-matrix systems

TL;DR

This paper develops a numerical large N method using a collective loop space framework to efficiently solve multi-matrix systems and compute their fluctuation spectra, overcoming previous complexity barriers.

Contribution

It introduces a constrained optimization scheme in loop space that handles large N multi-matrix problems with high precision and computational efficiency.

Findings

Successfully solves the general two matrix model problem.

Achieves solutions with close to 10^4 variables.

Provides accurate fluctuation spectra relevant for physical applications.

Abstract

In this work we revisit the problem of solving multi-matrix systems through numerical large $N$ methods. The framework is a collective, loop space representation which provides a constrained optimization problem, addressed through master-field minimization. This scheme applies both to multi-matrix integrals ($c=0$ systems) and multi-matrix quantum mechanics ($c=1$). The complete fluctuation spectrum is also computable in the above scheme, and is of immediate physical relevance in the later case. The complexity (and the growth of degrees of freedom) at large $N$ have stymied earlier attempts and in the present work we present significant improvements in this regard. The (constrained) minimization and spectrum calculations are easily achieved with close to $10^4$ variables, giving solution to Migdal-Makeenko, and collective field equations. Considering the large number of dynamical (loop) variables and the extreme nonlinearity of the problem, high precision is obtained when confronted with solvable cases. Through numerical results presented, we prove that our scheme solves, by numerical loop space methods, the general two matrix model problem.