Multiple Orthogonal Polynomials and Random Walks

TL;DR

This paper develops a framework connecting multiple orthogonal polynomials with dual Markov chains, analyzing their recurrence, transience, and steady states, with explicit results for Jacobi-Piñeiro polynomials and their associated random walks.

Contribution

It introduces a novel method to construct dual stochastic matrices from multiple orthogonal polynomials, extending classical Markov chain analysis to this new setting.

Findings

Explicit formula for type I Jacobi–Piñeiro polynomials

Parameter regions for recurrence and transience of the chains

Examples of recurrent and transient Jacobi–Piñeiro random walks

Abstract

Given a non-negative Jacobi matrix describing higher order recurrence relations for multiple orthogonal polynomials of type~II and corresponding linear forms of type I, a general strategy for constructing a pair of stochastic matrices, dual to each other, is provided. The corresponding Markov chains (or 1D random walks) allow, in one transition, to reach for the N-th previous states, to remain in the state or reach for the immediately next state. The dual Markov chains allow, in one transition, to reach for the N-th next states, to remain in the state or reach for immediately previous state. The connection between both dual Markov chains is discussed at the light of the Poincar\'e's theorem on ratio asymptotics for homogeneous linear recurrence relations and the Christoffel-Darboux formula within the sequence of multiple orthogonal polynomials and linear forms of type I. The Karlin-McGregor representation formula is extended to both dual random walks, and applied to the discussion of the corresponding generating functions and first-passage distributions. Recurrent or transient character of the Markov chain is discussed. Steady state and some conjectures on its existence and the relation with mass points are also given. The Jacobi-Pi\~neiro multiple orthogonal polynomials are taken as a case study of the described results. For the first time in the literature, an explicit formula for the type~I Jacobi--Pi\~neiro polynomials is determined. The region of parameters where the Markov chains are recurrent or transient is given, and it is conjectured that when recurrent, the Markov chains are null recurrent and, consequently, the expected return times are infinity. Examples of recurrent and transient Jacobi--Pi\~neiro random walks are constructed explicitly.