Return Probability of Quantum and Correlated Random Walks.

Area of Science:

• Quantum mechanics
• Statistical physics
• Probability theory

Background:

• Return probability is a fundamental concept in classical random walk analysis.
• Understanding return probabilities is crucial for characterizing the long-term behavior of random processes.

Purpose of the Study:

• To investigate the return probability of quantum random walks.
• To analyze the return probability of correlated random walks.
• To compare these probabilities on a one-dimensional integer lattice.

Main Methods:

• Employed the path counting method for analysis.
• Utilized mathematical expressions involving Legendre polynomials.
• Derived generating functions for quantum walk return probabilities.

Main Results:

• Demonstrated that return probabilities for both quantum and correlated random walks can be expressed using Legendre polynomials.
• Showed that the generating function for quantum walk return probability is related to elliptic integrals of the first and second kinds.

Conclusions:

• Established a novel connection between quantum/correlated random walks and special functions (Legendre polynomials, elliptic integrals).
• Provides a new analytical framework for studying quantum and correlated random walks on lattices.