Bounded Discrete Bridges

TL;DR

This paper corrects and extends previous work on the height distribution of bounded discrete bridges, providing a rigorous proof and exploring periodic walks with certain polynomial properties.

Contribution

It offers a corrected proof of the limiting distribution for bounded discrete bridges and investigates the case of periodic walks with specific polynomial decompositions.

Findings

The limiting height distribution is Rayleigh, consistent with earlier simulations.

The proof is valid within the domain where dominance properties hold.

Periodic walks with decomposable characteristic polynomials are analyzed.

Abstract

In 2010 Banderier and Nicodeme consider the height of bounded discrete bridges and conclude to a limiting Rayleigh distribution. This result is correct although their proof is partly erroneous. They make asymptotic simplifications based upon dominance properties of the roots of the kernel of the walk within a disk centered at the origin, but these dominance properties apply only upon a positive real segment. However the very good agreement of simulations with their asymptotic expansion of the probability distribution in case of {\L}ukasiewicz bridges let us think that their proof could be corrected. This is the scope of the present article which provides a proof using the dominance property only in its domain of validity. We also consider the case of periodic walks, a topic not considered in Banderier-Nicodeme2010. We limit ourselves to walks whose characteristic polynomial decomposes over $\bC$ without repeated factors.