Asymptotics of Laurent Polynomials of Odd Degree Orthogonal with Respect to Varying Exponential Weights

Abstract

Let Λ R denote the linear space over R spanned by z k , k ∈ Z. Define the real inner product (with varying exponential weights)Define the even degree and odd degree monic orthogonal Laurent polynomials: π π π 2n (z) := (ξ (2n) n ) −1 φ 2n (z) and π π π 2n+1 (z) := (ξ (2n+1) −n−1 ) −1 φ 2n+1 (z). Asymptotics in the double-scaling limit as N, n → ∞ such that N/n = 1+o(1) of π π π 2n+1 (z) (in the entire complex plane), ξ (2n+1) −n−1 , φ 2n+1 (z) (in the entire complex plane), and Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequencek∈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].

Asymptotics of Recurrence Relation Coefficients, Hankel Determinant Ratios, and Root Products Associated with Laurent Polynomials Orthogonal with Respect to Varying Exponential Weights

“…In [52,53], the L-polynomials were taken to be orthogonal with respect to the varying exponential measure (cf. Remark 1.…”

“…Define the bilinear form (with varying exponential weight) ·, · L as follows: ·, · L : R × R → R, (f, g) → f, g L := L(f (z)g(z)) = R f (s)g(s) exp(−N V (s)) d s, N ∈ N. It is a fact [7,40,43,60] (see, also, the proofs of Lemmata 2.4 and 2.2.2 in [52] and [53], respectively) that the bilinear form ·, · L thus defined is an inner product if, and only if, H (−2m) 2m > 0 and H (−2m) 2m+1 > 0 for m ∈ Z + 0 (see (1.8) …”

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

“…• the 'odd' equilibrium measure has compact support which consists of the disjoint union of a finite number of bounded real intervals; in fact, as shown in [51], supp(µ…”

“…are not arbitrary; rather, they satisfy an n-dependent and (locally) solvable system of 2(N +1) moment conditions (transcendental equations; see [51], Lemma 3.5);…”

“…Since the main results of this paper are asymptotics (as n → ∞) for π π π 2n (z) (z ∈ C), ξ (2n) n and φ 2n (z) (z ∈ C), which are, via Lemma 2.2.1, Equation (2.2), and Equations (1.2) and (1.4), related to RHP1 for e Y : C \ R → SL 2 (C), no further reference, henceforth, to RHP2 (and Lemma 2.2.2) for o Y : C \ R → SL 2 (C) will be made (see [51] for the complete details of the asymptotic analysis of RHP2). In the ensuing analysis, the large-n behaviour of the solution of RHP1 (see Lemma 2.2.1, Equation (2.2)), hence asymptotics for π π π 2n (z) (in the entire complex plane), ξ (2n) n and φ 2n (z) (in the entire complex plane), are extracted.…”