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Random walks in Weyl chambers

Denis Denisov, Will FitzGerald, Kaiyuan Zhang

TL;DR

The paper advances the theory of random walks in Weyl chambers of type $C$ and $D$ by establishing the existence of positive harmonic functions for walks killed on exit and deriving sharp tail asymptotics for exit times under near-optimal moment conditions, without relying on symmetry. The core approach blends diffusion-approximation error analysis with Green-function bounds, leveraging polynomial harmonic functions $h$ and Doob $h$-transforms to describe the conditioned process. A key contribution is showing that, for a homogeneous polynomial harmonic $h$ of degree $p$, the exit-time tail satisfies $ ext{P}( au_x>n)\sim olinebreak rac{ ext{const} imes V(x)}{n^{p/2}}$ and that the conditioned position converges to a Gaussian density weighted by $h$, yielding a polynomially tempered universality in these structured cones. The results extend existing Weyl-chamber analyses to the C and D types under weaker symmetry and optimal moment assumptions, with potential implications for related stochastic models and universality phenomena in growing systems and random matrices.

Abstract

We study a $d$-dimensional random walk with zero mean and finite variance in the Weyl chambers of type C and D. Under optimal moment assumptions we construct positive harmonic functions for random walks killed on exiting Weyl chambers. We also find the tail asymptotics for the exit time of the random walk from Weyl chambers.

Random walks in Weyl chambers

TL;DR

The paper advances the theory of random walks in Weyl chambers of type and by establishing the existence of positive harmonic functions for walks killed on exit and deriving sharp tail asymptotics for exit times under near-optimal moment conditions, without relying on symmetry. The core approach blends diffusion-approximation error analysis with Green-function bounds, leveraging polynomial harmonic functions and Doob -transforms to describe the conditioned process. A key contribution is showing that, for a homogeneous polynomial harmonic of degree , the exit-time tail satisfies and that the conditioned position converges to a Gaussian density weighted by , yielding a polynomially tempered universality in these structured cones. The results extend existing Weyl-chamber analyses to the C and D types under weaker symmetry and optimal moment assumptions, with potential implications for related stochastic models and universality phenomena in growing systems and random matrices.

Abstract

We study a -dimensional random walk with zero mean and finite variance in the Weyl chambers of type C and D. Under optimal moment assumptions we construct positive harmonic functions for random walks killed on exiting Weyl chambers. We also find the tail asymptotics for the exit time of the random walk from Weyl chambers.
Paper Structure (10 sections, 17 theorems, 164 equations)

This paper contains 10 sections, 17 theorems, 164 equations.

Table of Contents

  1. Introduction, assumptions and main results
  2. Introduction
  3. Background and notation for ordered random walks
  4. Assumptions and main results
  5. Preliminary estimates
  6. Estimate for the error term and the Green function of Brownian motion
  7. Construction of superharmonic function
  8. Construction of harmonic function
  9. Upper bound for $\mathbb{P}(\tau_x>n)$
  10. Proof of Theorem \ref{['thm.p.tau-poly']}.

Key Result

Theorem 4

Let Assumption ass-m-h hold. Then the function is finite, positive and harmonic for $(x+S(n))_{n \geq 0}$ killed on leaving $K$, that is

Theorems & Definitions (37)

  • Remark 2
  • Theorem 4
  • Remark 5
  • Theorem 6
  • Remark 7
  • Lemma 8
  • proof
  • Lemma 9
  • proof
  • Lemma 10
  • ...and 27 more