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

TL;DR

This paper analyzes the asymptotic behavior of non-diagonal multiple orthogonal polynomials with a cubic weight, revealing phase transitions and complex zero distributions through vector critical measures and Riemann-Hilbert analysis.

Contribution

It provides detailed asymptotics and phase transition descriptions for non-hermitian multiple orthogonal polynomials with a cubic weight, using vector critical measures and spectral curves.

Findings

Asymptotic zero distributions are characterized by vector critical measures.

Zeros of different polynomial types exhibit distinct asymptotic behaviors.

Phase transitions depend on the ratio of polynomial degrees, affecting the topology of zero distribution curves.

Abstract

We consider the type I multiple orthogonal polynomials (MOPs) $(A_{n,m}, B_{n,m})$ and type II MOPs $P_{n,m}$, satisfying non-hermitian orthogonality with respect to the weight $e^{-z^3}$ on two unbounded contours on $\mathbb C$. Under the assumption that $$ n,m \to \infty, \quad \frac{n}{n+m}\to \alpha \in (0, 1) $$ we find the detailed asymptotics of the MOPs, and describe the phase transitions of this limit behavior as a function of $\alpha$. This description is given in terms of vector critical measures, which are saddle points of an energy functional comprising both attracting and repelling forces. These critical measures are characterized by a cubic equation (spectral curve), and their components $\mu_j$ live on trajectories of a canonical quadratic differential $\varpi$ on the Riemann surface of this equation, which was object of study in our previous paper [Adv. Math. 302 (2016), 1137--1232]. The asymptotic zero distribution of the polynomials $A_{n,m}$ and $P_{n,m}$ are given by appropriate combinations of the components of the vector critical measure. However, in the case of the zeros of $B_{n,m}$ the behavior is totally different, and can be described in terms of the balayage of $\mu_2 - \mu_3$ onto certain curves on the plane. These curves are constructed with the aid of $\varpi$, and their topology has three very distinct characters, depending on the value of $\alpha$, and are obtained from the critical graph of $\varpi$. Once the trajectories and vector critical measures are studied, the main asymptotic technical tool is the analysis of a $3\times 3$ Riemann-Hilbert problem characterizing the MOPs. We illustrate our findings with results of several numerical experiments, and formulate some conjectures and empirical observations based on these experiments.