Matrix Bootstrap Approximation without Positivity Constraint

TL;DR

This paper introduces a novel bootstrap approximation method for the Hermitian one-matrix model that avoids positivity constraints, accurately reproducing known solutions and extending applicability to Minkowski models.

Contribution

The paper presents a new bootstrap framework that determines eigenvalue distributions and moments self-consistently without positivity constraints, applicable to both Euclidean and Minkowski one-matrix models.

Findings

Accurately reproduces exact solutions for Euclidean models

Matches perturbative results for Minkowski models

Eliminates the sign problem in the approximation process

Abstract

We propose a bootstrap approximation method for the Hermitian one-matrix model that does not rely on positivity constraints. The theoretical foundation of this method is that the one-matrix model admits an eigenvalue distribution $\rho(\lambda)$, and that the moments $w_n$ generated from it satisfy the loop equations. Our framework is designed to numerically determine a self-consistent pair of $\rho(\lambda)$ and $w_n$ that simultaneously satisfies these two requirements. In the concrete implementation the least-squares method is employed, and since the sign problem is absent in this formulation, the method can be formally applied to the Minkowski one-matrix model as well, provided that the one-cut structure of the resolvent is assumed. Actual numerical calculations show that this bootstrap approximation reproduces, with very high accuracy, the exact solutions for Euclidean-type models and the perturbative results for Minkowski-type models.