Discrete harmonic polynomials in multidimensional orthants

TL;DR

This paper investigates the relationship between discrete harmonic polynomials and the geometry of multidimensional cones, establishing connections with Coxeter groups and harmonic functions in random walks.

Contribution

It demonstrates that the existence of positive discrete harmonic polynomials implies certain geometric properties of cones and explores this relationship in dimensions two and higher.

Findings

Existence of harmonic polynomials implies geometric properties of cones.

In dimension two, harmonic polynomials coincide with probabilistic harmonic functions.

The work links algebraic, geometric, and probabilistic aspects of random walks in cones.

Abstract

We consider multidimensional random walks in pyramidal cones (or multidimensional orthants), which are intersections of a finite number of half-spaces. We explore the connection between the existence of (positive) discrete harmonic polynomials for the random walks, with Dirichlet conditions on the boundary of the cone, and geometric properties of the cone, being or not the Weyl chamber of a finite Coxeter group. We prove that the first property implies the second, derive the converse in dimension two and show in this case that it coincides with the probabilistic harmonic function.