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Discrete harmonic polynomials in multidimensional orthants

Emmanuel Humbert, Kilian Raschel

TL;DR

The paper analyzes discrete harmonic polynomials on multidimensional orthants for random walks with Dirichlet boundary conditions, revealing a sharp link between existence of ${a^+}$-polynomials and Weyl chambers of finite Coxeter groups. It proves that such a polynomial exists only when the cone is a Weyl chamber, and in dimension $2$ the converse holds, with the discrete harmonic polynomial intimately tied to the probabilistic harmonic function, the réduite, and the walk’s boundary behavior. A constructive covariance-prescription framework is developed to realize prescribed Weyl-chamber geometries via inventory polynomials, enabling explicit models with targeted reflection groups (e.g., $B_3$, $H_3$). The work further provides a practical 2D-to-3D construction to realize any admissible covariance matrix, along with explicit one-parameter families of zero-drift models sharing the same covariance, enriching the toolkit for studying random walks in cones and their harmonic structures.

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.

Discrete harmonic polynomials in multidimensional orthants

TL;DR

The paper analyzes discrete harmonic polynomials on multidimensional orthants for random walks with Dirichlet boundary conditions, revealing a sharp link between existence of -polynomials and Weyl chambers of finite Coxeter groups. It proves that such a polynomial exists only when the cone is a Weyl chamber, and in dimension the converse holds, with the discrete harmonic polynomial intimately tied to the probabilistic harmonic function, the réduite, and the walk’s boundary behavior. A constructive covariance-prescription framework is developed to realize prescribed Weyl-chamber geometries via inventory polynomials, enabling explicit models with targeted reflection groups (e.g., , ). The work further provides a practical 2D-to-3D construction to realize any admissible covariance matrix, along with explicit one-parameter families of zero-drift models sharing the same covariance, enriching the toolkit for studying random walks in cones and their harmonic structures.

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.

Paper Structure

This paper contains 9 sections, 12 theorems, 76 equations, 1 figure.

Table of Contents

  1. Introduction and main results
  2. Proof of Theorem \ref{['harmonic0']}
  3. Proof of Theorem \ref{['P0harm']}
  4. Proof of Theorem \ref{['comparingfP']}
  5. Prescription of the covariance matrix
  6. A construction
  7. Proof of Theorem \ref{['cov_presc']}
  8. Examples
  9. One-parameter families

Key Result

Theorem 1

If there exists an ${\text{a}}^+$-polynomial $P$ on $C$, then $P$ is unique and $C$ is a Weyl chamber of a finite Coxeter group. If $d=2$ and $C$ is a Weyl chamber of a finite Coxeter group (i.e., if the interior angle of $C$ has the form $\pi/n$ for some positive integer $n$), then there exists an

Figures (1)

  • Figure 1: Illustration of Theorem \ref{['harmonic']}. Leftmost model: the function $P(i,j)=ij$ is harmonic for the simple random walk (with uniform probabilities $\frac{1}{4}$). Second model (called tandem walk), with uniform probabilities $\frac{1}{3}$: the polynomial $P(i,j)=ij(i+j)$ is discrete harmonic. Third model, with probabilities $a(1,-1)=\frac{1}{2}$ and the other probabilities are $\frac{1}{6}$: the previous function is also harmonic. Rightmost model (with uniform probabilities $\frac{1}{4}$): the function $P(i,j)=ij(i+j)(i+2j)$ is discrete harmonic.

Theorems & Definitions (24)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 14 more