Communications on pure and applied mathematics vol:53 issue:6 pages:736-785
The equilibrium measure in the presence of an external field plays a role in a number of areas in analysis, for example, in random matrix theory: The limiting mean density of eigenvalues is precisely the density of the equilibrium measure. Typical behavior for the equilibrium measure is: 1. it is positive on the interior of a finite number of intervals, 2. it vanishes like a square root at endpoints, and 3. outside the support, there is strict inequality in the Euler-Lagrange variational conditions. If these conditions hold, then the limiting local eigenvalue statistics is loosely described by a "bulk," in which there is universal behavior involving the sine kernel, and "edge effects," in which there is a universal behavior involving the Airy kernel. Through techniques from potential theory and integrable systems, we show that this "regular" behavior is generic for equilibrium measures associated with real analytic external fields. In particular, we show that for any one-parameter family of external fields V/c, the equilibrium measure exhibits this regular behavior except for an at most countable number of values of c. We discuss applications of our results to random matrices, orthogonal polynomials, and integrable systems.