Global asymptotic expansions of the Laguerre polynomials—a Riemann–Hilbert approach

Abstract

By using the steepest descent method for Riemann-Hilbert problems introduced by Deift-Zhou (Ann Math 137:295-370, 1993), we derive two asymptotic expansions for the scaled Laguerre polynomial L (α) n (νz) as n → ∞, where ν = 4n + 2α + 2. One expansion holds uniformly in a right halfplane Re z ≥ δ 1 , 0 < δ 1 < 1, which contains the critical point z = 1; the other expansion holds uniformly in a left half-plane Re z ≤ 1 − δ 2 , 0 < δ 2 < 1 − δ 1 , which contains the other critical point z = 0. The two half-planes together cover the entire complex z-plane. The critical points z = 1 and z = 0 correspond, respectively, to the turning point and the singularity of the differential equation satisfied by L (α) n (νz).

Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

“…The parametrix can be constructed, out of the Airy function and its derivative, as in Section 6.1 below, and in [41, (3.74)]; see also [10,13,35]. …”

On the condition number of the critically-scaled Laguerre Unitary Ensemble

“…These asymptotics are derived via the Deift-Zhou method of nonlinear steepest descent as applied to orthogonal polynomials [4] (see also [3] for an introduction). For fixed α, this problem was addressed by Vanlessen [27] for the generalized weight x α e −Q(x) dx (see also [21] for Q(x) = x). From the classical work of Szegö [24] and [27], the asymptotic expansion of L N (x) as N → ∞ near x = 0 is given in terms of Bessel functions.…”

Uniform asymptotics for orthogonal polynomials with exponential weight on the positive real axis

“…It has been successfully applied to obtain globally uniform asymptotic expansions of several families of orthogonal polynomials, see e.g. [13,[17][18][19].…”

“…First, for the classical Laguerre case (Q(x) = x), the results are contained in Szegő's seminal book [15]. Their global asymptotic expansion is derived by Qiu and Wong [13]. See also [2,8] for the case α < −1, where the Laguerre polynomials are no longer orthogonal on the positive real axis, but on a curve in the complex plane.…”