Matrix Bootstrap Approximation without Positivity Constraint

TL;DR

This work develops a positivity-free bootstrap for the large-$N$ Hermitian one-matrix model by directly utilizing the eigenvalue density $ρ_E(λ)$ and the moments $w_n$, enforcing self-consistency through loop equations via a finite-degree polynomial approximation $ρ_E^{(P)}(λ)$ and a least-squares fit. The approach replaces the conventional positivity constraints with a self-consistency framework, and is extended to Minkowski-type models through a density-matrix (master-field) interpretation that leads to a contour $Γ$-supported density $ρ_M(z)$ and a single-cut ansatz. Numerical results show that the method reproduces exact Euclidean solutions with high accuracy and perturbative Minkowski results, while providing consistent, though more challenging, behavior for Minkowski runs and endpoint-handling strategies (endpoint-free vs endpoint-fixed). The work demonstrates a scalable, fast alternative to SDP-based bootstrap in the large-$N$ limit and offers a foundation for applying similar ideas to more complex multi-matrix systems such as the IKKT model, with future work on regularization, branch structure, and explicit positivity checks as consistency tests. Overall, the paper delivers a practical, algebraically grounded framework that leverages $ρ_E(λ)$ and $w_n$ self-consistency to study large-$N$ matrix models in both Euclidean and Minkowski signatures.

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.

Figures (3)

• Figure 1: In the two plots, the exact eigenvalue distributions $\rho_{E}(\lambda)$ are compared with its polynomial approximations $\rho_{E}^{(P)}(\lambda)$. The upper panel corresponds to $g=-1$, while the lower panel corresponds to $g=0.05$. The dashed lines represent $\rho_{E}(\lambda)$, and the solid lines represent $\rho_{E}^{(P)}(\lambda)$. In the polynomial approximation, the integration interval is fixed to $[-a,a]$, and the value of $\pm a$ obtained from the bootstrap approximation are indicated by cross marks. For both $g=-1$ and $g=0.05$, one finds that $\rho_{E}^{(P)}(\pm a)\neq 0$, and in particular for $g=0.05$ there is a significant discrepancy from the exact solution $\rho_{E}(\pm a)$.
• Figure 2: These figures show $\rho_{E}^{(P)}(\lambda)$ obtained by imposing the condition $\rho_{E}^{(P)}(\pm a)=0$ in \ref{['eq:new approximation anzats']}. In this case, a high level of accuracy is achieved for both $g=-1$ and $g=0.05$. As expected, the condition $\rho_{E}^{(P)}(\pm a)=0$ is indeed satisfied at the endpoints $\pm a$, which are indicated by cross marks.
• Figure 3: Plots of the real and imaginary parts of the formal solution $\rho_M(z)$ and its polynomial approximation $\rho_M^{(P)}(z)$ for $g=-1$ and $g=-0.3$. Each of $\rho_M(z)$ and $\rho_M^{(P)}(z)$ is defined as a complex-valued function supported on $\Gamma$ and $\Gamma^{(P)}$, respectively. By parametrizing them as $\rho_M(e^{i\theta}\lambda)$ and $\rho_M^{(P)}(e^{i\theta^{(P)}}\lambda)$, the two distributions are plotted on the same $\lambda$--$\rho$ plane. Note, however, that as a consequence of this parametrization, the difference between the angles $\theta$ and $\theta^{(P)}$ cannot be read off from this figure (however, as can be seen from Table \ref{['tab:minkowski_EF']}, the numerical difference between these quantities is in fact extremely small).