Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields

The equilibrium measure in the presence of an external field plays a role i n a number of areas in analysis, for example, in random matrix theory: The limiting mean density of eigenvalues is precisely the density of the equili brium 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 va riational conditions. If these conditions hold, then the limiting local eigenvalue statistics is loosely described by a "bulk," in which there is universal behavior involvi ng the sine kernel, and "edge effects," in which there is a universal behav ior involving the Airy kernel. Through techniques from potential theory and integrable systems, we show that this "regular" behavior is generic for eq uilibrium measures associated with real analytic external fields. In partic ular, 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 c ountable number of values of c. We discuss applications of our results to r andom matrices, orthogonal polynomials, and integrable systems. (C) 2000 Jo hn Wiley & Sons, Inc.