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Enumeration of walks in multidimensional orthants and reflection groups

Léa Gohier, Emmanuel Humbert, Kilian Raschel

TL;DR

This work connects the enumeration of walks in multidimensional orthants to reflection/Coxeter-group theory via a universal covariance-based domain, enabling two principal advances: (i) a structural link between the combinatorial group $G$ and a reflection group $H$ through a surjective morphism, yielding practical infiniteness criteria for $G$; and (ii) a spectral approach to excursion asymptotics through the principal Dirichlet eigenvalue $\lambda_1$ on a polyhedral nodal domain, with a complete classification of such domains in low dimensions. The core technical engine is the transformation to the domain $T=\Delta^{-1/2}\mathbb{R}_+^d$, the construction of the Jacobian-based morphism $J$, and the identification of $U$ as chamber intersections of finite Coxeter groups, which together determine both the algebraic structure of $G$ and the asymptotic exponent $\alpha=1+\sqrt{\lambda_1+(d/2-1)^2}$. The paper provides concrete criteria to decide when $G$ is infinite, tools to transfer information between $G$ and $H$, and a thorough spectral-combinatorial framework that yields explicit $\lambda_1$ and $\alpha$ in small dimensions, with three illustrative examples. This yields new exact asymptotics for walk models and broadens the applicability of Coxeter-group methods to high-dimensional lattice-path problems.

Abstract

We consider (random) walks in a multidimensional orthant. Using the idea of universality in probability theory, one can associate a unique polyhedral domain to any given walk model. We use this connection to prove two sets of new results. First, we are interested in a group of transformations naturally associated with any small step model; as it turns out, this group is central to the classification of walk models. We show a strong connection between this group and the reflection group through the walls of the polyhedral domain. As a consequence, we can derive various conditions for the combinatorial group to be infinite. Secondly, we consider the asymptotics of the number of excursions, whose critical exponent is known to be computable in terms of the eigenvalue of the above polyhedral domain. We prove new results from spectral theory on the eigenvalues of polyhedral nodal domains. We believe that these results are interesting in their own right; they can also be used to find new exact asymptotic results for walk models corresponding to these nodal polyhedral domains.

Enumeration of walks in multidimensional orthants and reflection groups

TL;DR

This work connects the enumeration of walks in multidimensional orthants to reflection/Coxeter-group theory via a universal covariance-based domain, enabling two principal advances: (i) a structural link between the combinatorial group and a reflection group through a surjective morphism, yielding practical infiniteness criteria for ; and (ii) a spectral approach to excursion asymptotics through the principal Dirichlet eigenvalue on a polyhedral nodal domain, with a complete classification of such domains in low dimensions. The core technical engine is the transformation to the domain , the construction of the Jacobian-based morphism , and the identification of as chamber intersections of finite Coxeter groups, which together determine both the algebraic structure of and the asymptotic exponent . The paper provides concrete criteria to decide when is infinite, tools to transfer information between and , and a thorough spectral-combinatorial framework that yields explicit and in small dimensions, with three illustrative examples. This yields new exact asymptotics for walk models and broadens the applicability of Coxeter-group methods to high-dimensional lattice-path problems.

Abstract

We consider (random) walks in a multidimensional orthant. Using the idea of universality in probability theory, one can associate a unique polyhedral domain to any given walk model. We use this connection to prove two sets of new results. First, we are interested in a group of transformations naturally associated with any small step model; as it turns out, this group is central to the classification of walk models. We show a strong connection between this group and the reflection group through the walls of the polyhedral domain. As a consequence, we can derive various conditions for the combinatorial group to be infinite. Secondly, we consider the asymptotics of the number of excursions, whose critical exponent is known to be computable in terms of the eigenvalue of the above polyhedral domain. We prove new results from spectral theory on the eigenvalues of polyhedral nodal domains. We believe that these results are interesting in their own right; they can also be used to find new exact asymptotic results for walk models corresponding to these nodal polyhedral domains.
Paper Structure (28 sections, 20 theorems, 88 equations, 7 figures)

This paper contains 28 sections, 20 theorems, 88 equations, 7 figures.

Table of Contents

  1. Introduction and main results
  2. The combinatorial group as a reflection group
  3. Preliminary notations and results
  4. Assumptions on the step set
  5. The group of the model
  6. Probabilistic estimates
  7. Four random walks
  8. A surjective morphism from the combinatorial group to a reflection group
  9. Preliminary results
  10. Main result
  11. Illustration in dimension two
  12. Applications
  13. Infinite group criterion in any dimension
  14. A method to prove that the reflection group is infinite
  15. A tool to determine H when G is known
  16. ...and 13 more sections

Key Result

Theorem 1

Let $\mathcal{S}$ be a step set satisfying it:hypothesis_irreducible, and let $\chi_\mathcal{S}$ be its inventory eq:inventory. The system of equations admits a unique solution in $(0,\infty)^d$, denoted by $\boldsymbol{x_0}$. Define the covariance matrix Let $\Delta^{-\frac{1}{2}}$ denote the inverse of the symmetric, positive definite square root of the covariance matrix $\Delta$, see eq:def_p

Figures (7)

  • Figure 1: Walks in orthants $\mathbb R_+^d$ in small dimensions $d=1,2,3$
  • Figure 2: The orthant $\mathbb R_+^d$ is mapped into the polyhedral domain (or pyramid) \ref{['eq:def_new_domain']}, which depends on the model (illustration in dimension two and three). In the new domain, the model has a covariance matrix equal to the identity.
  • Figure 3: Measure of the angle $\widehat{H_i H_j}$ between two hyperplanes $H_i$ and $H_j$ in $T$, in terms of the angle $\widehat{\left(u_i,u_j\right)}$ between the normal vectors $u_i$ and $u_j$ in \ref{['eq:def_u_i']}. The relationship between the angle $\widehat{H_i H_j}$ and $\widehat{\left(u_i,u_j\right)}$ is determined by the orientation of the normal vectors $u_i$ and $u_j$ being inward: $\widehat{\left(u_i,u_j\right)}=\pi - \widehat{H_i H_j}$.
  • Figure 4: In dimension $2$, the vectors $u_i$ and the hyperplanes $H_i$ are easily computed in terms of the correlation factor $a$. The chamber becomes the domain between the two hyperplanes $H_1$ and $H_2$, which is a wedge of opening $\arccos -a$, as already noticed in DeWa-15. See also Figure \ref{['fig:action_Delta']}.
  • Figure 5: Left picture: the model considered in Example \ref{['ex:ex2']}. Second picture: the model of Example \ref{['ex:ex3']}. Right picture: the model presented on Example \ref{['ex:ex4']}.
  • ...and 2 more figures

Theorems & Definitions (45)

  • Theorem 1: DeWa-15DeWa-19
  • Lemma 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Theorem 5
  • proof
  • Corollary 6
  • ...and 35 more