scholarly journals The Maximum of a Symmetric Next Neighbor Walk on the Nonnegative Integers

2014 ◽  
Vol 51 (1) ◽  
pp. 162-173
Author(s):  
Ora E. Percus ◽  
Jerome K. Percus
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Generating Functions ◽  
Two Dimensional ◽  
Symmetric Random Walk ◽  
One Dimensional ◽  
Dimensional Distribution

We consider a one-dimensional discrete symmetric random walk with a reflecting boundary at the origin. Generating functions are found for the two-dimensional probability distribution P{Sn = x, max1≤j≤nSn = a} of being at position x after n steps, while the maximal location that the walker has achieved during these n steps is a. We also obtain the familiar (marginal) one-dimensional distribution for Sn = x, but more importantly that for max1≤j≤nSj = a asymptotically at fixed a2 / n. We are able to compute and compare the expectations and variances of the two one-dimensional distributions, finding that they have qualitatively similar forms, but differ quantitatively in the anticipated fashion.

scholarly journals The Maximum of a Symmetric Next Neighbor Walk on the Nonnegative Integers

2014 ◽  
Vol 51 (01) ◽  
pp. 162-173
Author(s):  
Ora E. Percus ◽  
Jerome K. Percus
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Generating Functions ◽  
Two Dimensional ◽  
Symmetric Random Walk ◽  
One Dimensional ◽  
Dimensional Distribution

We consider a one-dimensional discrete symmetric random walk with a reflecting boundary at the origin. Generating functions are found for the two-dimensional probability distribution P{S n = x, max1≤j≤n S n = a} of being at position x after n steps, while the maximal location that the walker has achieved during these n steps is a. We also obtain the familiar (marginal) one-dimensional distribution for S n = x, but more importantly that for max1≤j≤n S j = a asymptotically at fixed a 2 / n. We are able to compute and compare the expectations and variances of the two one-dimensional distributions, finding that they have qualitatively similar forms, but differ quantitatively in the anticipated fashion.

scholarly journals Quantum walks and elliptic integrals

2010 ◽  
Vol 20 (6) ◽  
pp. 1091-1098 ◽  
Author(s):  
NORIO KONNO
Keyword(s):  
Random Walk ◽  
Generating Function ◽  
Elliptic Integral ◽  
Quantum Walk ◽  
Quantum Walks ◽  
Elliptic Integrals ◽  
Two Dimensional ◽  
One Dimensional ◽  
Similar Expression ◽  
Return Probability

Pólya showed in his 1921 paper that the generating function of the return probability for a two-dimensional random walk can be written in terms of an elliptic integral. In this paper we present a similar expression for a one-dimensional quantum walk.

Transition probability matrices for correlated random walks

1980 ◽  
Vol 17 (01) ◽  
pp. 253-258 ◽  
Author(s):  
R. B. Nain ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Difference Equations ◽  
Generating Functions ◽  
Transition Probability ◽  
Correlated Random Walk ◽  
One Dimensional ◽  
Correlated Random Walks ◽  
Transition Probability Matrices ◽  
Probability Matrices

For correlated random walks a method of transition probability matrices as an alternative to the much-used methods of probability generating functions and difference equations has been investigated in this paper. To illustrate the use of transition probability matrices for computing the various probabilities for correlated random walks, the transition probability matrices for restricted/unrestricted one-dimensional correlated random walk have been defined and used to obtain some of the probabilities.

On Certain Probability Models in Random Walk

1972 ◽  
Vol 21 (1-2) ◽  
pp. 71-76 ◽  
Author(s):  
Kanwar Sen ◽  
Y. L. Ahuja
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
Two Dimensional ◽  
Probability Models ◽  
Unit Step

This paper considers a two dimensional random walk in which a particle moves from its position a unit step either horizontally, or vertically, or diagonally forward or stay there with respective probabilities p, q, r and s ( p + q + r + s= 1). The probabilities that initially at (0, 0}, the position of the particle is ( x, y) at time t+ 0 after having in the line y = x, ( i) h crosses and (ii) g hits and b slips, have been determined alongwith their generating functions.

LATTICE GAS WAVE PROPAGATION MODELS INCLUDING DISSIPATION

1995 ◽  
Vol 03 (01) ◽  
pp. 69-93 ◽  
Author(s):  
YASUSHI SUDO ◽  
VICTOR W. SPARROW
Keyword(s):  
Wave Propagation ◽  
Random Walk ◽  
Diffusion Equation ◽  
Lattice Gas ◽  
Sound Propagation ◽  
Two Dimensional ◽  
Propagation Models ◽  
One Dimensional ◽  
Lattice Gas Models ◽  
Wave Models

New lattice gas models for one-dimensional (1D) and two-dimensional (2D) sound propagation have been recently proposed by the authors. These models were dissipationless and deterministic. In this paper, it will be shown how dissipation effects can be included into these lattice gas wave models. To simulate these dissipation effects, the lattice gas particles are assumed to take a random walk. The resulting models combine the authors' lattice gas wave models with published lattice gas models for the diffusion equation. The formulations are stable and consistent.

A modified random walk in the presence of partially reflecting barriers

1976 ◽  
Vol 13 (1) ◽  
pp. 169-175 ◽  
Author(s):  
Saroj Dua ◽  
Shobha Khadilkar ◽  
Kanwar Sen
Keyword(s):  
Random Walk ◽  
Generating Functions ◽  
One Dimensional ◽  
Unit Step ◽  
The One ◽  
The Right ◽  
Reflecting Barriers

The paper deals with the one-dimensional modified random walk in the presence of partially reflecting barriers at a and –b (a, b > 0). The simple one-dimensional random walk on a line is the motion-record of a particle which may extend over (–∞, + ∞) or be restricted to a portion of it by absorbing and/or reflecting barriers. Here we introduce the possibility of a particle staying put along with its moving a unit step to the right or to the left and find the bivariate generating functions of the probabilities of a particle reaching m (0 <m <a) under different conditions.

Two models of quantum random walk

Open Physics ◽  
2003 ◽  
Vol 1 (4) ◽  
Author(s):  
Jozef Košík
Keyword(s):  
Random Walk ◽  
Probability Distribution ◽  
Upper Bound ◽  
Explicit Solution ◽  
New Method ◽  
Speed Of Convergence ◽  
Quantum Random Walk ◽  
Dimensional Lattice ◽  
Special Model ◽  
One Dimensional

AbstractWe 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.

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