Analytic and numerical bootstrap for one-matrix model and “unsolvable” two-matrix model

A bstract We propose the relaxation bootstrap method for numerical solution of multi-matrix models in large N limit, developing and improving recent proposal H. Lin. It gives rigorous inequalities on single trace moments matrices up to a given “cutoff” order (length) moments. The combines usual loop equations positivity constraint correlation matrix have proof applicability this case one-matrix model where condition saddle point appears be equivalent presence supports eigenvalue distribution only real axis with positive weight. demonstrate efficiency our by solving analytically “unsolvable” two-matrix tr[A , B] 2 interaction quartic potentials, even solutions spontaneously broken discrete symmetry. region values computed allowed quickly shrinks increase cutoff, allowing precision about 6 digits generic couplings ? symmetric solutions. Our data are checked against known analytic results particular parameters.