Asymptotics for the number of walks in a Weyl chamber of type B

TL;DR

This paper derives asymptotic formulas for the number of lattice walks confined to a Weyl chamber of type B, with applications to tangled diagrams and vicious walkers, extending known results to more general cases.

Contribution

It provides the first comprehensive asymptotic formulas for walk counts in Weyl chambers of type B, covering fixed and free endpoints, and applies these to combinatorial models.

Findings

Asymptotic formulas for walk counts in Weyl chambers of type B.

Asymptotics for k-non-crossing tangled diagrams.

Asymptotics for k-vicious walkers in wall-restricted models.

Abstract

We consider lattice walks in $\R^k$ confined to the region $0<x_1<x_2...<x_k$ with fixed (but arbitrary) starting and end points. The walks are required to be "reflectable", that is, we assume that the number of paths can be counted using the reflection principle. The main results are asymptotic formulas for the total number of walks of length $n$ with either a fixed or a free end point for a general class of walks as $n$ tends to infinity. As applications, we find the asymptotics for the number of $k$-non-crossing tangled diagrams on the set $\{1,2,...,n\}$ as $n$ tends to infinity, and asymptotics for the number of $k$-vicious walkers subject to a wall restriction in the random turns model as well as in the lock step model. Asymptotics for all of these objects were either known only for certain special cases, or have only been partially determined or were completely unknown.