Variational ansatz for gaussian + Yang-Mills two matrix model compared with Monte-Carlo simulations in 't Hooft limit

In recent work, we have developed a variational principle for large N multi-matrix models based on the extremization of non-commutative entropy. Here, we test the simplest variational ansatz for our entropic variational principle with Monte-Carlo measurements. In particular, we study the two matrix model with action Tr[{m^2 \over 2} (A_1^2 + A_2^2) - {1 \over 4} [A_1,A_2]^2] which has not been exactly solved. We estimate the expectation values of traces of products of matrices and also those of traces of products of exponentials of matrices (Wilson loop operators). These are compared with a Monte-Carlo simulation. We find that the simplest wignerian variational ansatz provides a remarkably good estimate for observables when \(m^2\) is of order unity or more. For small values of m^2 the wignerian ansatz is not a good approximation: the measured correlations grow without bound, reflecting the non-convergence of matrix integrals defining the pure commutator squared action. Comparison of this ansatz with the exact solution of a two matrix model studied by Mehta is also summarized. Here the wignerian ansatz is a good approximation both for strong and weak coupling.