Large Parameter Behavior of Equilibrium Measures

We study the equilibrium measure for a logarithmic potential in the presence of an external field V∗(ξ) + t p(ξ), where t is a parameter, V∗(ξ) is a smooth function and p(ξ) a monic polynomial. When p(ξ) is of an odd degree, the equilibrium measure is shown to be supported on a single interval as |t| is sufficiently large. When p(ξ) is of an even degree, the equilibrium measure is supported on two disjoint intervals as t is negatively large; it is supported on a single interval for convex p(x) as t is positively large and is likely to be supported on multiple disjoint intervals for non-convex p(x). The support of the equilibrium measure shrinks to isolated points as |t| → +∞ in all the cases that we consider. For sufficiently large |t|, each topological component of the support contains a local minimizing point of the external field V∗(ξ) + t p(ξ); a “potential well” phenomenon.