The δ Steepest Descent Method and the Asymptotic Behavior of Polynomials Orthogonal on the Unit Circle with Fixed and Exponentially Varying Nonanalytic Weights

摘要

The steepest descent method for asymptotic analysis of matrix Riemann-Hilbert problems was introduced by Deift and Zhou in 1993 [14]. A matrix Riemann-Hilbert problem is specified by giving a triple (∑, v, N) consisting of an oriented contour L in the complex z-plane, a matrix function v : ∑ -> SL(N) which is usually taken to be continuous except at self-intersection points of I where a certain compatibility condition is required, and a normalization condition N as z -> ∞. If I is not bounded, certain asymptotic conditions are required of v in order to have compatibility with the normalization condition. Consider an analytic function M : C ∑ -> SL(IM) taking continuous boundary values M+ (z) (resp., M_(z)) on I from the left (resp., right). The Riemann-Hilbert problem (∑,v,N) is then to find such a matrix M.(z) satisfying the normalization condition X as z -> ∞ and the jump condition M+(z) = M_(z)v(z) whenever z is a non-self-intersection point of I (so the left and right boundary values are indeed well defined). The steepest descent method of Deift and Zhou applies to certain Riemann-Hilbert problems where the jump matrix v(z) depends on an auxiliary control parameter, and is a method for extracting asymptotic properties of the solution M(z) (and indeed proving the existence anduniqueness of solutions along the way) when the control parameter tends to a singular limit of interest.