Return probability of quantum and correlated random walks

TL;DR

This paper investigates the return probabilities of quantum and correlated random walks on a one-dimensional lattice, expressing these probabilities through Legendre polynomials and elliptic integrals, advancing understanding of their fundamental behaviors.

Contribution

It introduces a novel analytical approach linking return probabilities to special functions for quantum and correlated random walks.

Findings

Return probability expressed via Legendre polynomial

Generating function involves elliptic integrals

Provides explicit formulas for quantum and correlated walks

Abstract

The analysis of the return probability is one of the most essential and fundamental topics in the study of classical random walks. In this paper, we study the return probability of quantum and correlated random walks in the one-dimensional integer lattice by the path counting method. We show that the return probability of both quantum and correlated random walks can be expressed in terms of the Legendre polynomial. Moreover, the generating function of the return probability can be written in terms of elliptic integrals of the first and second kinds for the quantum walk.