K. T. R. McLaughlin

Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle

Strong asymptotics of polynomials orthogonal on the unit circle with respect to a weight of the form

A Riemann-Hilbert approach to asymptotic questions for orthogonal polynomials

Abstract A few years ago the authors introduced a new approach to study asymptotic questions for orthogonal polynomials. In this paper we give an overview of our method and review the results which have been obtained in Deift et al. (Internat. Math. Res. Notices (1997) 759, Comm. Pure Appl. Math. 52 (1999) 1491, 1335), Deift (Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert Approach, Courant Lecture Notes, Vol. 3, New York University, 1999), Kriecherbauer and McLaughlin (Internat. Math. Res. Notices (1999) 299) and Baik et al. (J. Amer. Math. Soc. 12 (1999) 1119). We mainly consider orthogonal polynomials with respect to weights on the real line which are either (1) Freud-type weights d α(x)= e −Q(x) d x (Q polynomial or Q(x)=|x|β, β>0), or (2) varying weights d α n (x)= e −nV(x) d x (V analytic, lim |x|→∞ |V(x)|/ log |x|=∞ ). We obtain Plancherel–Rotach-type asymptotics in the entire complex plane as well as asymptotic formulae with error estimates for the leading coefficients, for the recurrence coefficients, and for the zeros of the orthogonal polynomials. Our proof starts from an observation of Fokas et al. (Comm. Math. Phys. 142 (1991) 313) that the orthogonal polynomials can be determined as solutions of certain matrix valued Riemann–Hilbert problems. We analyze the Riemann–Hilbert problems by a steepest descent type method introduced by Deift and Zhou (Ann. Math. 137 (1993) 295) and further developed in Deift and Zhou (Comm. Pure Appl. Math. 48 (1995) 277) and Deift et al. (Proc. Nat. Acad. Sci. USA 95 (1998) 450). A crucial step in our analysis is the use of the well-known equilibrium measure which describes the asymptotic distribution of the zeros of the orthogonal polynomials.

Asymptotics of Laurent polynomials of even degree orthogonal with respect to varying exponential weights

Let Λ denote the linear space over R spanned by z, k ∈Z. Define the real inner product (with varying exponential weights) 〈·, ·〉L : Λ ×ΛR→R, ( f , g) 7→ ∫ R f (s)g(s) exp(−N V(s)) ds, N∈N, where the external field V satisfies: (i) V is real analytic onR\ {0}; (ii) lim|x|→∞(V(x)/ ln(x2+1))=+∞; and (iii) lim|x|→0(V(x)/ ln(x−2+1))=+∞. Orthogonalisation of the (ordered) base {1, z−1, z, z−2, z2, . . . , z−k, z, . . . } with respect to 〈·, ·〉L yields the even degree and odd degree orthonormal Laurent polynomials {φm(z)}m=0:φ2n(z)=ξ (2n) −n z −n+· · ·+ξ n z, ξ n >0, andφ2n+1(z)=ξ −n−1 z−n−1+· · ·+ξ (2n+1) n z , ξ −n−1 >0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π2n(z) := (ξ (2n) n )φ2n(z) and π2n+1(z) := (ξ (2n+1) −n−1 ) φ2n+1(z). Asymptotics in the double-scaling limit as N,n→ ∞ such that N/n=1+o(1) of π2n+1(z) (in the entire complex plane), ξ (2n+1) −n−1 , φ2n+1(z) (in the entire complex plane), and Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequence { ck= ∫ R s exp(−N V(s)) ds } k∈Z are obtained by formulating the odd degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on R, and then extracting the largen behaviour by applying the non-linear steepest-descent method introduced in [1] and further developed in [2, 3]. 2000 Mathematics Subject Classification. (Primary) 30E20, 30E25, 42C05, 45E05, 47B36: (Secondary) 30C15, 30C70, 30E05, 30E10, 31A99, 41A20, 41A21, 41A60 Abbreviated Title. Asymptotics of Odd Degree Orthogonal Laurent Polynomials