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Journal of Mathematical Physics

Transitional probabilities for the four-state random walk on a lattice in the presence of partially reflecting boundaries

Ramakrishna Janaswamy

Published May 1, 2009• 1 citation
Publisher

Abstract

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.

DOI
10.1063/1.3122768

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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.
A closed form is given for a three-parameter family of transition matrices describing continuous-time nonsymmetric random walks on a d=1d=1d=1 semi-infinite lattice. The boundary conditions are general, allowing for an arbitrary mixture of reflection, trapping, and sojourn. Special cases give the quantum propagator for the half-lattice, and transition matrices for the random walk on the Bethe lattice and the single-server queue $\frac{\frac{M}{M}}{\frac{1}{\ensuremath{\infty}}}$.
In this paper, we focus on discrete-time random walks (DTRWs) on the path graph with reflecting walls. Also we consider the corresponding continuous-time random walks (CTRWs), discrete-time quantum walks (DTQWs) and continuous-time quantum walks (CTQWs). We give formulas for the return probabilities of these four walks (DTRWs, CTRWs, DTQWs, CTQWs) at the boundary point of the path graph and discuss their relationships. Furthermore, by considering the Ehrenfest model, we derive the scaling limit theorems for the return probabilities of both the CTRW and the CTQW.
Here, a new two-dimensional process, discrete in time and space, that yields the results of both a random walk and a quantum random walk, is introduced. This model describes the population distribution of four coin states |1>,-|1>, |0> -|0> in space without interference, instead of two coin states |1>, |0> .For the case of no boundary conditions, the model is similar to a Markov chain with a stochastic matrix, i.e., it conserves the population distribution of the four coin states, and by using a proper transformation, yield probability distributions of the two quantum states |1>, |0> in space, similar to a unitary operator. Numerical results for a quantum random walk on infinite and finite lines are introduced.
Quantum percolation describes the problem of a quantum particle moving through a disordered system. While certain similarities to classical percolation exist, the quantum case has additional complexity due to the possibility of Anderson localisation. Here, we consider a directed discrete-time quantum walk as a model to study quantum percolation of a two-state particle on a two-dimensional lattice. Using numerical analysis we determine the fraction of connected edges required (transition point) in the lattice for the two-state particle to percolate with finite (non-zero) probability for three fundamental lattice geometries, finite square lattice, honeycomb lattice, and nanotube structure and show that it tends towards unity for increasing lattice sizes. To support the numerical results we also use a continuum approximation to analytically derive the expression for the percolation probability for the case of the square lattice and show that it agrees with the numerically obtained results for the discrete case. Beyond the fundamental interest to understand the dynamics of a two-state particle on a lattice (network) with disconnected vertices, our study has the potential to shed light on the transport dynamics in various quantum condensed matter systems and the construction of quantum information processing and communication protocols.
Quantum walks are known to have nontrivial interactions with absorbing boundaries. In particular it has been shown that an absorbing boundary in the one-dimensional quantum walk partially reflects information, as observed by absorption probability computations. In this paper, we shift our sights from the local phenomena of absorption probabilities to the global behavior of finite absorbing quantum walks in one dimension. We conduct our analysis by approximating the eigenbasis of the associated absorbing quantum walk operator matrix Qn{Q}_{n}Qn​, where nnn is the lattice size. The conditional probability distributions of these finite absorbing quantum walks exhibit distinct behavior at various timescales, namely wavelike reflections for times t=O(n)t=O(n)t=O(n), fractional quantum revivals for t=O(n2)t=O({n}^{2})t=O(n2), and stability for t=O(n3)t=O({n}^{3})t=O(n3). At the end of this paper, we demonstrate the existence of fractional quantum revivals in other sufficiently regular quantum walk systems.