Persistent random walk on a site-disordered one-dimensional lattice: Photon subdiffusion

The investigation of random walks is central to a variety of stochastic processes in physics, chemistry and biology. To describe a transport phenomenon, we study a variant of the one-dimensional persistent random walk, which we call a zero-one-only process. It makes a step in the same direction as the previous step with probability p, and stops to change the direction with 1 − p. By using the generating-function method, we calculate its characteristic quantities such as the statistical moments and probability of the first return.

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Phase transition in one-dimensional random walk with partially reflecting boundaries

Advances in Applied Probability ◽

10.1017/s000186780001524x ◽

1985 ◽

Vol 17 (03) ◽

pp. 594-606 ◽

Cited By ~ 8

Author(s):

Ora E. Percus

Keyword(s):

Random Walk ◽

Transition Probability ◽

Dimensional Lattice ◽

One Dimensional ◽

Boundary Case ◽

Reflecting Boundaries ◽

The Mean ◽

The One ◽

The Relationship ◽

Asymptotic Forms

We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p). The probability distribution (Pn (m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of P n(m) result, depending on the relationship between transition probability and the reflection probability.

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Phase transition in one-dimensional random walk with partially reflecting boundaries

Advances in Applied Probability ◽

10.2307/1427121 ◽

1985 ◽

Vol 17 (3) ◽

pp. 594-606 ◽

Cited By ~ 9

Author(s):

Ora E. Percus

Keyword(s):

Random Walk ◽

Transition Probability ◽

Dimensional Lattice ◽

One Dimensional ◽

Boundary Case ◽

Reflecting Boundaries ◽

The Mean ◽

The One ◽

The Relationship ◽

Asymptotic Forms

We consider an asymmetric random walk, with one or two boundaries, on a one-dimensional lattice. At the boundaries, the walker is either absorbed (with probability 1–ρ) or reflects back to the system (with probability p).The probability distribution (Pn(m)) of being at position m after n steps is obtained, as well as the mean number of steps before absorption. In the one-boundary case, several qualitatively different asymptotic forms of Pn(m) result, depending on the relationship between transition probability and the reflection probability.

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Two models of quantum random walk

Open Physics ◽

10.2478/bf02475903 ◽

2003 ◽

Vol 1 (4) ◽

Cited By ~ 3

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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