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Journal of Physics A Mathematical and Theoretical

Transitional probabilities for the 4-state random walk on a lattice

Ramakrishna Janaswamy

Published Apr 2, 2008• 9 citations
Publisher

Abstract

The diffusion and Schrödinger propagators have been known to coexist on a lattice when a particle undergoing random walk is endowed with two states of spin in addition to the two states of direction in a 1+1 spacetime dimension. In this paper we derive explicit expressions for the various transitional probabilities by employing generating functions and transform methods. The transitional probabilities are all expressed in terms of a one-dimensional integral involving trigonometric functions and/or Chebyshev polynomials of the first and second kind from which the spacetime continuum limits of the diffusion equation and Schrödinger equation follow directly.

DOI
10.1088/1751-8113/41/15/155306

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The diffusion and Schrödinger propagators have been known to coexist on a lattice when a particle undergoing random walk is endowed with two states of spin in addition to the two states of direction in a 1+1 spacetime dimension. In this paper we derive explicit expressions for the various transitional probabilities by employing generating functions and transform methods. The transitional probabilities are all expressed in terms of a one-dimensional integral involving trigonometric functions and/or Chebyshev polynomials of the first and second kind from which the spacetime continuum limits of the diffusion equation and Schrödinger equation follow directly.
The four-state random walk (4RW) model, wherein the particle is endowed with two states of spin and two states of directional motion in each space coordinate, permits a stochastic solution of the Schrödinger equation (or the equivalent parabolic equation) without resorting to the usual analytical continuation in complex space of the particle trajectories. Analytical expressions are derived here for the various transitional probabilities in a 4RW by employing generating functions and eigenfunction expansions when the particle moves on a 1+1 space-time lattice with two-point boundary conditions. The most general case of dissimilar boundaries with partially reflecting boundary conditions is treated in this paper. The transitional probabilities are all expressed in terms of a finite summation involving trigonometric functions and/or Chebyshev polynomials of the second kind that are characteristics of diffusion and Schrödinger equations, respectively, in the 4RW model. Results for the special case of perfectly absorbing boundaries are compared to numerical values obtained by directly counting paths in the random walk simulations.
The Schrödinger and diffusion equations are normally related only through a formal analytic continuation. There are apparently no intermediary partial differential equations with physical interpretations that can form a conceptual bridge between the two. However, if one starts off with a symmetric binary random walk on a lattice then it is possible to show that both equations occur as approximate descriptions of different aspects of the same classical probabilistic system. This suggests that lattice calculations may prove to be a useful intermediary between classical and quantum physics.
Abstract We present an overview of two models of quantum random walk. In the first model, the discrete quantum random walk, we present the explicit solution for the recurring amplitude of the quantum random walk on a one-dimensional lattice. We also introduce a new method of solving the problem of random walk in the most general case and use it to derive the hitting amplitude for quantum random walk on the hypercube. The second is a special model based on a local interaction between neighboring spin-1/2 particles on a one-dimensional lattice. We present explicit results for the relevant quantities and obtain an upper bound on the speed of convergence to limiting probability distribution.
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.
We analyze several families of one and two-dimensional nearest neighbor Quantum Random Walks. Using a multivariate generating function analysis we give a simplified proof of a known phenomenon for two-chirality walks on the line, namely that the walk has linear speed rather than the diffusive behavior observed in classical random walks. We also demonstrate Airy phenomena between the regions of polynomial and exponential decay. For a three-chirality walk on the line we demonstrate similar behavior, with the addition of a bound state, in which the probability of finding the particle at the origin does not go to zero with time. For each of these walks on the line we obtain exact formulae for the leading asymptotic term of the wave function and the location probabilities. Analyzing two-dimensional walks we again find a region of polynomial decay which grows linearly with time. The limiting shape of the feasible region is, however, quite different. The limit region turns out to be an algebraic set, which we characterize as the rational image of a compact algebraic variety. We also compute the probability profile within the limit region, which is essentially a negative power of the Gaussian curvature of the same algebraic variety. We close with preliminary work concerning walks in higher dimensions.