Uniqueness and convergence for a non-convex two-matrix model

We study the planar Schwinger–Dyson equations of the Hermitian two-matrix model with potential V = ½(A²+B²) + (g/4)(A⁴+B⁴) − (h/4)[A,B]², which is not convex for h > 0. We prove that they have exactly one tracial solution, and that the empirical distribution of the matrix model converges almost surely to it, for all 0 ≤ h ≤ g ≤ 1, for g = 1 and 0 ≤ h ≤ 1.007, and in an explicit region of the (g,h) plane for every g > 0. Two estimates combine. First, the moments of (A+iB)*(A+iB) satisfy a recursion, derived from the Schwinger–Dyson equations with a sharp inequality for the quartic term, which bounds ‖A+iB‖ for every solution. Second, a sum-of-squares identity shows that for h ≤ g the Hessian of V is at least 1 − (53/200)h‖A+iB‖², improving the constant 1/2 that follows from ‖[A,B]‖ ≤ ½‖A+iB‖². So V is uniformly convex on a convex set that contains every solution, and two solutions coupled through stationary free Langevin dynamics contract. With results of Guionnet and Maurel-Segala on trapping potentials this gives uniqueness and convergence. The identity is printed in full and checked in exact arithmetic by a short program. Without it, the same argument proves the result for 0 ≤ h ≤ 0.537 at g = 1 with no computer assistance.