Multiple Orthogonal Polynomials and Random Walks

Abstract

A. 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 Poincaré's theorem on ratio asymptotics for homogeneous linear recurrence relations and the Christo el-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-Piñeiro 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-Piñeiro polynomials is determined. Then, the region of parameters where the Jacobi matrix is non-negative is given. Moreover, two stochastic matrices, describing two dual random walks, one allowing to reach for the two previous states, remain or reach for the next, and the other allowing to reach for the two next states, remain or reach for the previous, are given in terms of the Jacobi matrix and the values of the multiple orthogonal polynomials of type II and corresponding linear forms of type I at the point 1.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-Piñeiro random walks are constructed explicitly.

Classical and quantum walks on paths associated with exceptional Krawtchouk polynomials

“…Karlin and McGregor themselves [13] and others [14,15,16,17] used multivariate orthogonal polynomials to analyze multidimensional or composition BDP. For applications of matrix orthogonal polynomials to quasi-birth and death processes one may consult [18] or [19,20] in the discrete time case and again for discrete time, random walks that multiple polynomials entail have been looked at recently in [21].…”

Hypergeometric Multiple Orthogonal Polynomials and Random Walks

“…In this paper we continue our investigations on random walk orthogonal polynomials of multiple type. In our previous paper [15] we found that the ideas of Karlin and McGregor [33], see also [31,32], can be extended from standard orthogonality to the multiple orthogonality scenario, whenever the Jacobi matrix is nonnegative and bounded. Now, instead of birth and death Markov chains, in which the non zero transition probabilities could only happen for near neighbors, we have that transitions to the 𝑁-th previous states are permitted.…”

“…In [15] we presented the Jacobi-Piñeiro multiple orthogonal polynomials as a case study, and several properties were given. In particular, we showed the region where the Jacobi-Piñeiro random walks were recurrent or transient in terms of the Jacobi-Piñeiro parameters.…”

“…The large 𝑛 limit of the corresponding Jacobi matrix leads to a particular interesting case with a Toeplitz matrix describing the transitions. At the time we finished the writing of [15] a new paper appeared [39], in where new multiple orthogonal polynomials based on Gauss hypergeometric function weights were given. The corresponding Jacobi matrix is bounded and nonnegative.…”

“…For that aim we explicitly compute the value at unity of the type I linear forms. Then, whenever the zeros of the type II multiple orthogonal polynomials are confined in the interior of the support of the weight system, the type II multiple orthogonal polynomials at unity are also positive and our method in [15] is applicable. Notice that this confinement of zeros happens for example for algebraic Chebyshev (AT) systems of weights, in particular for Nikishin systems as the hypergeometric system of weights is in some region of parameters, see [39].…”