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Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. (English) Zbl 0944.42013
In this important and interesting paper, the authors again illustrate the awesome power of the Deift-Zhou steepest descent method associated with Riemann-Hilbert problems. Let \(V: \mathbb{R}\to\mathbb{R}\) be real-valued and analytic on \(\mathbb{R}\), with \[ \lim_{|x|\to\infty} V(x)/\log(1+ x^2)=\infty. \] For \(n\geq 1\), let \(\{p_k(x; n)\}^\infty_{k= 0}\) denote the sequence of orthonormal polynomials for the weight \(\exp(-nV)\), so that \[ \int p_k(x; n)p_j(x; n)\exp(- nV(x)) dx= \delta_{jk}\quad\forall j,k\geq 0. \] The authors derive Plancherel-Rotach asymptotics for \(p_n(z; n)\) as \(n\to\infty\), valid in every region of the plane. The precision of the asymptotics on and off the real line is remarkable. The proofs involve the Fokas-Its-Kitaev identify for the orthonormal polynomials as solutions of a Riemann-Hilbert problem, followed by the Deift-Zhou steepest descent technique. The details are intricate, but are clearly presented, and the main ideas are summarized to guide the reader through the proofs.
As an application, the authors prove universality limits that arise in random matrix theory. These involve the weighted reproducing kernel \[ K_n(x,y)= e^{-{n\over 2}(V(x)+ V(y))} \sum^{n- 1}_{j= 0} p_j(x; n)p_j(y; n) \] and have the form \[ {1\over n\psi(a)} K_n\Biggl(a+ {s\over n\psi(a)}, a+{t\over n\psi(a)}\Biggr)= {\sin\pi(s- t)\over \pi(s- t)}+ O\Biggl({1\over n}\Biggr), \] uniformly for \(s\), \(t\) in compact subsets of \(\mathbb{R}\). Here \(a\) lies in a subinterval of the support of the equilibrium measure \(\mu_V\) associated with the field \(V\), and \(\psi\) is the density of that equilibrium measure.

MSC:
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
15B52 Random matrices (algebraic aspects)
41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
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