Matrix Bootstrap Approximation without Positivity Constraint

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 $ρ(λ)$, 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 $ρ(λ)$ and $w_n$ that simultaneously satisfies these two requirements. In the concrete implementation, we employ a least-squares method, for which no sign problem arises in principle, and therefore the method can be formally applied also to Minkowski-type models. 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.