Analytic trajectory bootstrap for matrix models

TL;DR

This work develops and applies an analytic trajectory bootstrap for large-$N$ matrix models with a $ ext{tr}[A,B]^2$ interaction, bypassing positivity relaxations by leveraging nonlinear loop equations and a carefully constructed singularity ansatz for generating functions. By organizing observables into one- and multi-length trajectories and enforcing square-root branch structures, the authors obtain highly accurate solutions for both the one-matrix and two-matrix models, including detailed eigenvalue densities and predictions for long-words observables up to $L_{ m max}=22$. The method yields rigorous convergence toward accurate values for key observables such as $igraket{ ext{Tr}A^2}$, $igraket{ ext{Tr}A^4}$, and mixed moments, and also exposes symmetry-breaking two-cut solutions with nonzero order parameters. The results align with, and in some cases surpass, existing positivity bounds, and point to broad applicability to more complex multi-matrix and SUSY theories, with potential implications for lattice gauge theory and Wilson-loop analyses in the large-length regime.

Abstract

We revisit the large $N$ two-matrix model with $\text{tr}[A,B]^2$ interaction and quartic potentials by the analytic trajectory bootstrap, where $A$ and $B$ represent the two matrices. In the large $N$ limit, we can focus on the single trace moments associated with the words composed of the letters $A$ and $B$. Analytic continuations in the lengths of the words and subwords lead to analytic trajectories of single trace moments and intriguing intersections of different trajectories. Inspired by the one-cut solutions of one-matrix models, we propose some simple ansatzes for the singularity structure of the two-matrix generating functions and the corresponding single trace moments. Together with the self-consistent constraints from the loop equations, we determine the free parameters in the ansatzes and obtain highly accurate solutions for the two-matrix model at a low computational cost. For a given length cutoff $L_\text{max}$, our results are within and more accurate than the positivity bounds from the relaxation method, such as about 6-digit accuracy for $L_\text{max}=18$. The convergence pattern suggests that we achieve about $8$-digit accuracy for $L_\text{max}=22$. As the singularity structure is closely related to the eigenvalue distributions, we further present the results for various types of eigenvalue densities. In the end, we study the symmetry breaking solutions using more complicated ansatzes.

Figures (9)

• Figure 1: The analytic trajectories associated with the Green's functions $\left\langle\text{Tr}\,A^{n-L_\mathcal{O}}\mathcal{O} \right\rangle$ of the two-matrix model \ref{['2mm-definition']}. Here $n$ is the length of the full word $A^{n-L_\mathcal{O}}\mathcal{O}$, and $L_\mathcal{O}$ is the length of the subword $\mathcal{O}$. The trajectories are labelled by the powers of the letters as in \ref{['G2k']}. According to the contraction limit \ref{['contraction']} and symmetries of the Green's functions, these analytic trajectories exhibit intriguing intersection phenomena, which lead to nontrivial matching conditions for the one-length analytic ansatz \ref{['1-length-ansatz']}. We write down some explicit words associated with the intersection points. For clarity, we have stripped off the common factor $(1+(-1)^n)/2$ associated with $\mathbb Z_2$ symmetry.
• Figure 2: The one-length eigenvalue density $\rho^{(1)}(z)$ of the two-matrix model \ref{['2mm-definition']} with $g=h=1$, which encodes the information of the one-length Green's functions $\left\langle\text{Tr}A^{n}\right\rangle$. We use \ref{['rho1-rho2']}, \ref{['2mm-rho-expansion']} and the one-cut solutions at $L_\text{max}=6,10,14,18,22$ to evaluate $\rho^{(1)}(z)$. As our results converge rapidly, the curves for $L_\text{max}=14,18,22$ are indistinguishable from each other. The eigenvalue distribution is consistent with and more accurate than the Monte Carlo results in Jha:2021exo.
• Figure 3: The relative error of the eigenvalue density $\delta\rho^{(1)}=\rho^{(1)}/\rho^{(1)}_\text{ref}-1$ for various $L_\text{max}$. We choose the $L_\text{max}=22$ solution as the reference density $\rho^{(1)}_\text{ref}$. As $L_\text{max}$ increases, the approximate density improves uniformly on the support of the eigenvalue distribution.
• Figure 4: The generalized one-length eigenvalue densities $\rho_{\mathcal{O}}^{(1)}(z)$ of the two-matrix model \ref{['2mm-definition']} with $g=h=1$, which are associated with the generalized one-length Green's functions $\left\langle\text{Tr}\,A^{n}\mathcal{O}\right\rangle$, where $\mathcal{O}=B^0, B^2, \dots, B^{18}$. We evaluate them using \ref{['2mm-rho-expansion']}, \ref{['rho1-rho2-generalized']} and the one-cut solution at $L_\text{max}=22$.
• Figure 5: The two-length eigenvalue density $\rho^{(2)}(z_1,z_2)$ of the two-matrix model \ref{['2mm-definition']} with $g=h=1$. The density of eigenvalues is related to the two-length Green's functions $\left\langle\text{Tr}A^{n_1}B^{n_2}\right\rangle$. We use \ref{['2mm-rho-expansion']} and the 1-cut solution at $L_\text{max}=22$ to evaluate $\rho^{(2)}(z_1,z_2)$.