  We study the equilibrium measure for a logarithmic potential in the presence
of an external field V*(x) + tp(x), where t is a parameter, V*(x) is a smooth
function and p(x) a monic polynomial. When p(x) is of an odd degree, the
equilibrium measure is shown to be supported on a single interval as |t| is
sufficiently large. When p(x) 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).
