Analytic trajectory bootstrap for matrix models

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.