Two Stochastic Models Of a Random Walk In The U(n)-Spherical Duals Of U(n + 1)

TL;DR

This paper introduces two stochastic models for a random walk in the U(n)-spherical duals of U(n+1), utilizing matrix-valued orthogonal polynomials and spherical functions, with applications to urn and Young diagram models.

Contribution

It presents two novel stochastic models based on matrix-valued orthogonal polynomials and spherical functions for random walks in U(n)-spherical duals of U(n+1).

Findings

Developed an urn model for the random walk.

Constructed a Young diagram model for the same walk.

Connected the models to matrix-valued orthogonal polynomials.

Abstract

The random walk to be considered takes place in the d- spherical dual of the group U(n + 1), for a fixed finite dimensional irreducible representation d of U(n). The transition matrix comes from the three term recursion relation satisfied by a sequence of matrix valued orthogonal polynomials built up from the irreducible spherical functions of type d of SU(n + 1). One of the stochastic models is an urn model and the other is a Young diagram model.