TY - JOUR

T1 - Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory

AU - Deift, P.

AU - Kriecherbauer, T.

AU - McLaughlin, K. T.R.

AU - Venakides, S.

AU - Zhou, X.

PY - 1999/11

Y1 - 1999/11

N2 - We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e-nV(x)dx on the line as n → ∞. 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 recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [19, 20]. The Riemann-Hilbert problem is analyzed in turn using the steepest-descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμv for V as analyzed in [8].

AB - We consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn(x)dx = e-nV(x)dx on the line as n → ∞. 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 recently in [31] and also in [4]. An additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials (see also [4]). The orthogonal polynomial problem is formulated as a Riemann-Hilbert problem following [19, 20]. The Riemann-Hilbert problem is analyzed in turn using the steepest-descent method introduced in [12] and further developed in [11, 13]. A critical role in our method is played by the equilibrium measure dμv for V as analyzed in [8].

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U2 - 10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1

DO - 10.1002/(SICI)1097-0312(199911)52:11<1335::AID-CPA1>3.0.CO;2-1

M3 - Article

AN - SCOPUS:0033440723

SN - 0010-3640

VL - 52

SP - 1335

EP - 1425

JO - Communications on Pure and Applied Mathematics

JF - Communications on Pure and Applied Mathematics

IS - 11

ER -