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Generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields. (English) Zbl 1022.31001
An external field is a continuous function $$V: \mathbb R\to\mathbb R$$ such that $$V(x)/\log |x|\to +\infty$$ as $$|x|\to +\infty$$. The weighted energy of a Borel probability measure $$\mu$$ is given by $I_V(\mu)=-\iint\log |s-t|d\mu (s) d\mu (t)+2\int V(t) d\mu (t),$ and the extremal weighted energy is $$E_V=\inf I_V(\mu)$$, where the infimum is taken over all such (nonnegative) measures. There is a unique Borel probability measure $$\mu_V$$, called the equilibrium measure, such that $$I_V(\mu_V)=E_V$$. Let $$S_V$$ denote the support of $$\mu_V$$, which is necessarily compact. It is known that if $$V$$ is real analytic, then $$\mu_V$$ has a density $$\psi_V$$ and $$S_V$$ is a finite union of closed intervals. If $$V$$ is an external field, then there is a real constant $$\ell_V$$ such that $$\int \log |x-t|d\mu (t)-V(x)\leq \ell_V$$ for all real $$x$$ with equality for each $$x$$ in $$S_V$$. A real analytic external field $$V$$ is called regular if (i) the above inequality is strict for each $$x$$ in $$\mathbb R\backslash S_V$$, (ii) $$\psi_V >0$$ on the interior of $$S_V$$, (iii) $$\psi_V$$ vanishes like a square root at each endpoint of $$S_V$$.
A large part of the paper is devoted to the study of the family $$\{V/c: c>0\}$$, where $$V$$ is a given real analytic external field. It is shown, for example, that (i)$$V/c$$ is regular for all but a discrete set of values of $$c$$, (ii) $$V/c$$ is regular for all sufficiently small values of $$c$$, and for such $$c$$ the support $$S_{V/c}$$ is a union of intervals each containing precisely one point at which $$V$$ attains its global minimum. It is also shown that regularity is a generic property: regular external fields form an open dense subset of the set of real analytic external fields equipped with a suitable metric.
Applications to random matrices, orthogonal polynomials, and integrable systems are discussed.

##### MSC:
 31A15 Potentials and capacity, harmonic measure, extremal length and related notions in two dimensions 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 82B10 Quantum equilibrium statistical mechanics (general)
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