Critical measures for vector energy: Asymptotics of non-diagonal multiple orthogonal polynomials for a cubic weight

Abstract

We consider the type I multiple orthogonal polynomials (MOPs) (An,m, Bn,m), deg An,m ≤ n − 1, deg Bn,m ≤ m − 1, and type II MOPs Pn,m, deg Pn,m = n + m, satisfying non-hermitian orthogonality with respect to the weight e −z 3 on two unbounded contours γ 1 and γ 2 on C, with (in the case of type II MOPs) n conditions on γ 1 and m on γ 2 . Under the assumption that n, m → ∞, n n + m → α ∈ (0, 1)we find the detailed (rescaled) asymptotics of An,m, Bn,m and Pn,m on C, and describe the phase transitions of this limit behavior as a function of α. This description is given in terms of the vector critical measure µα = (µ 1 , µ 2 , µ 3 ), the saddle point of an energy functional comprising both attracting and repelling forces. This critical measure is characterized by a cubic equation (spectral curve), and its components µ j live on trajectories of a canonical quadratic differential on the Riemann surface of this equation. The structure of these trajectories and their deformations as functions of α was object of study in our previous paper [Adv. Math. 302 (2016), 1137-1232, and some of the present results strongly rely on the analysis carried out there.We conclude that the asymptotic zero distribution of the polynomials An,m and Pn,m are given by appropriate combinations of the components µ j of the vector critical measure µα. However, in the case of the zeros of Bn,m the behavior is totally different, and can be described in terms of the balayage of the signed measure µ 2 − µ 3 onto certain curves on the plane. These curves are constructed with the aid of the canonical quadratic differential , and their topology has three very distinct characters, depending on the value of α, and are obtained from the critical graph of .Once the trajectories and vector critical measures are studied, the main asymptotic technical tool is the Deift-Zhou nonlinear steepest descent analysis of a 3 × 3 matrix-valued Riemann-Hilbert problem characterizing both (An,m, Bn,m) and Pn,m, which allows us to obtain the limit behavior of both types of MOPs simultaneously.We illustrate our findings with results of several numerical experiments, explain the computational methodology, and formulate some conjectures and empirical observations based on these experiments.

Global‐phase portrait and large‐degree asymptotics for the Kissing polynomials

“…We see that this distinction between the one and two cut regimes will also play a fundamental role in the present analysis, as hinted at by Figure 2. This potential‐theoretic approach, known now as the Gonchar–Rakhmanov–Stahl (GRS) program, has been carried out in various scenarios, and we refer the reader to many excellent works on the subject 24,31–37 …”

“…This potential-theoretic approach, known now as the Gonchar-Rakhmanov-Stahl (GRS) program, has been carried out in various scenarios, and we refer the reader to many excellent works on the subject. 24,[31][32][33][34][35][36][37] Despite many successful applications of potential theory to the analysis of non-Hermitian orthogonal polynomials via the GRS program, we adopt an alternate viewpoint based on deformation techniques born from advances in the theory of random matrices and integrable systems. We will make heavy use of the technique known as continuation in parameter space, first developed in the context of integrable systems (cf.…”

Lemniscate ensembles with spectral singularity

“…The strong asymptotics of p c j,N were extensively studied in [12,13,15,34], see also recent works [35,36] on the case with multiple point charges. We also refer the reader to [18,[28][29][30][31]37] for the strong asymptotics of planar orthogonal polynomials associated with some other classes of potentials.…”

Generalised Airy polynomials

“…where C a contour in the complex plane, were investigated in [14,23,31], while the semiclassical weight ω(x; t) = exp − 1 3 x 3 + tx , x ∈ C (1.6) with t ∈ R and C is a contour in the complex plane, was discussed in [2,3,4,9,13,14,22]. These studies of the weights (1.5) and (1.6) are for contours in the complex plane.…”