Critical curve of two-matrix models $ABBA$, $A\{B,A\}B$ and $ABAB$, Part I: Monte Carlo

For a family of two-matrix models $$\frac{1}{2}\mathrm{Tr}(A^2+B^2)-\frac{g}{4}\mathrm{Tr}(A^4+B^4)-\{\begin{array}{l}\frac{h}{2}\mathrm{Tr}(ABAB)\\ \frac{h}{4}\mathrm{Tr}(ABAB+ABBA)\\ \frac{h}{2}\mathrm{Tr}(ABBA)\end{array}$$ with hermitian $A$ and $B$, we provide, in each case, a Monte Carlo estimate of the boundary of the maximal convergence domain in the $(h,g)$-plane. The results are discussed comparing with exact solutions (in agreement with the only analytically solved case) and phase diagrams obtained by means of the functional renormalization group.