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Valence independent formula for the equilibrium measure

We derive a new formula for the equilibrium measure for eigenvalues of random matrices sampled from polynomial perturbations of the GUE, valid in the one-cut case. The virtue of our formula is that it depends on the potential only implicitly through the endpoints of support of the equilibrium measure. Our motivation is the problem of computing explicit formulas for generating functions which enumerate graphs embedded in a Riemann surface. To demonstrate the utility of our formula for the equilibrium measure, we derive a formula for the generating function $e_1$ enumerating maps on the torus. This formula is "valence independent" in the sense that it holds regardless of what numbers of edges are allowed to meet at vertices; furthermore it subsumes formulas for $e_1$ given by other authors as special cases.