Asymptotics of Recurrence Relation Coefficients, Hankel Determinant Ratios, and Root Products Associated with Laurent Polynomials 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 〈⋅,⋅〉 _L :Λ^ℝ×Λ^ℝ→ℝ, $(f,g)\mapsto \int_{\mathbb{R}}f(s)g(s)\exp (-{N}V(s))\mathop {d}s$ , N∈ℕ, where V satisfies: (i) V is real analytic on ℝ∖{0}; (ii) lim  _{| x |→∞}(V(x)/ln (x ²+1))=+∞; and (iii) lim  _{| x |→0}(V(x)/ln (x ⁻²+1))=+∞. Orthogonalisation of the (ordered) base $\lbrace 1,z^{-1},z,z^{-2},z^{2},\ldots ,z^{-k},z^{k},\ldots \rbrace$ with respect to 〈⋅,⋅〉 _L yields the even degree and odd degree orthonormal Laurent polynomials (OLPs) $\lbrace \phi_{m}(z)\rbrace_{m=0}^{\infty}$ : φ _2n (z)=∑ ⁿ _k=−n ξ ⁽²ⁿ⁾ _k z ^k, ξ ⁽²ⁿ⁾ _n >0, and φ _2n+1(z)=∑ ⁿ _k=−n−1 ξ ⁽²ⁿ⁺¹⁾ _k z ^k, ξ ⁽²ⁿ⁺¹⁾_−n−1 >0. Associated with the even degree and odd degree OLPs are the following two pairs of recurrence relations: z φ _2n (z)=c ^♯_2n φ _2n−2(z)+b ^♯_2n φ _2n−1(z)+a ^♯_2n φ _2n (z)+b ^♯_2n+1 φ _2n+1(z)+c ^♯_2n+2 φ _2n+2(z) and z φ _2n+1(z)=b ^♯_2n+1 φ _2n (z)+a ^♯_2n+1 φ _2n+1(z)+b ^♯_2n+2 φ _2n+2(z), where c ^♯₀ =b ^♯₀ =0, and c ^♯_2k >0, k∈ℕ, and z ⁻¹ φ _2n+1(z)=γ ^♯_2n+1 φ _2n−1(z)+β ^♯_2n+1 φ _2n (z)+α ^♯_2n+1 φ _2n+1(z)+β ^♯_2n+2 φ _2n+2(z)+γ ^♯_2n+3 φ _2n+3(z) and z ⁻¹ φ _2n (z)=β ^♯_2n φ _2n−1(z)+α ^♯_2n φ _2n (z)+β ^♯_2n+1 φ _2n+1(z), where β ^♯₀ =γ ^♯₁ =0, β ^♯₁ >0, and γ ^♯_2l+1 >0, l∈ℕ. Asymptotics in the double-scaling limit N,n→∞ such that N/n=1+o(1) of the coefficients of these two pairs of recurrence relations, Hankel determinant ratios associated with the real-valued, bi-infinite strong moment sequence $\lbrace c_{k}=\int_{\mathbb{R}}s^{k}\exp (-{N}V(s))\mathop {d}s\rbrace_{k\in \mathbb{Z}}$ , and the products of the (real) roots of the OLPs are obtained by formulating the even degree and odd degree OLP problems as matrix Riemann-Hilbert problems on ℝ, and then extracting the large-n behaviours by applying the non-linear steepest-descent method introduced in (Ann. Math. 137(2):295–368, [1993]) and further developed in (Commun. Pure Appl. Math. 48(3):277–337, [1995]) and (Int. Math. Res. Not. 6:285–299, [1997]).