Analysis of Bidirectional Ballot Sequences and Random Walks Ending in their Maximum

TL;DR

This paper investigates the properties of admissible lattice paths ending at their maximum, revealing their connection to Chebyshev polynomials and proving a conjecture on ballot sequences through bijections.

Contribution

It establishes a link between admissible paths and Chebyshev polynomials and proves Zhao's conjecture using a novel bijection between random walks and binary sequences.

Findings

Probability of admissible paths related to Chebyshev polynomials

Asymptotic analysis of path parameters like length and height

Proof of Zhao's conjecture on ballot sequences

Abstract

Consider non-negative lattice paths ending at their maximum height, which will be called admissible paths. We show that the probability for a lattice path to be admissible is related to the Chebyshev polynomials of the first or second kind, depending on whether the lattice path is defined with a reflective barrier or not. Parameters like the number of admissible paths with given length or the expected height are analyzed asymptotically. Additionally, we use a bijection between admissible random walks and special binary sequences to prove a recent conjecture by Zhao on ballot sequences.