Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory
P. Deift et al., Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory, COM PA MATH, 52(11), 1999, pp. 1335-1425
We consider asymptotics for orthogonal polynomials with respect to varying
exponential weights w(n)(x)dx = e(-nV(x))dx on the line as n --> infinity.
The potentials V are assumed to be real analytic, with sufficient growth at
infinity. The principle results concern Plancherel-Rotach-type asymptotics
for the orthogonal polynomials down to the axis. Using these asymptotics,
we then prove universality for a variety of statistical quantities arising
in the theory of random matrix models, some of which have been considered r
ecently in [31] and also in [4]. An additional application concerns the asy
mptotics of the recurrence coefficients and leading coefficients for the or
thonormal polynomials (see also [4]).
The orthogonal polynomial problem is formulated as a Riemann-Hilbert proble
m following [19, 20]. The Riemann-Hilbert problem is analyzed in turn using
the steepest-descent method introduced in [12] and further developed in [1
1, 13]. A critical role in our method is played by the equilibrium measure
d(mu V) for V as analyzed in [8]. (C) 1999 John Wiley & Sons, Inc.