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

Let Λℝ denote the linear space over ℝ spanned by zk, k ∈ ℤ. Define the real inner product (with varying exponential weights) 〈·,·〉ℒ:Λℝ × Λℝ → ℝ, (f,g) ↦ ∫ℝ f(s)g(s) exp(−NV(s))ds, N ∈ ℕ, where the external field V satisfies the following: (i) V is real analytic on ℝ \ {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,zk,…} with respect to 〈·,·〉ℒ yields the even degree and odd degree orthonormal Laurent polynomials {Φm(z)}m=0∞:Φ2n(z)=ξ−n(2n)z−n+…+ξn(2n)zn,ξn(2n)>0, and Φ2n+1(z)=ξ−n−1(2n+1)z−n−1+…+ξn(2n+1)zn, ξ−n−1(2n+1)>0. Define the even degree and odd degree monic orthogonal Laurent polynomials: π2n(z):=(ξn(2n))−1Φ2n(z) and π2n+1(z):=(ξ−n−1(2n+1))−1Φ2n+1(z). Asymptotics in the double-scaling limit as N, n → ∞ such that N/n = 1 + o(1) of π2n(z) (in the entire complex plane), ξn(2n), Φ2n(z) (in the entire complex plane), and Hankel determinant ratios associated with the real-valued, bi-infinite, strong moment sequence {ck = ∫ℝsk exp(−NV(s))ds}k∈ℤ are obtained by formulating the even degree monic orthogonal Laurent polynomial problem as a matrix Riemann-Hilbert problem on ℝ, and then extracting the large-n behaviour by applying the nonlinear steepest-descent method introduced by P. Deift and X. Zhou and further developed by P. Deift, S. Venakides, and X. Zhou.