Topic
Heterogeneous random walk in one dimension
About: Heterogeneous random walk in one dimension is a research topic. Over the lifetime, 2812 publications have been published within this topic receiving 76343 citations.
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01 Dec 1984TL;DR: The goal will be to interpret Polya’s beautiful theorem that a random walker on an infinite street network in d-dimensional space is bound to return to the starting point when d = 2, but has a positive probability of escaping to infinity without returning to the Starting Point when d ≥ 3, and to prove the theorem using techniques from classical electrical theory.
Abstract: Probability theory, like much of mathematics, is indebted to physics as a source of problems and intuition for solving these problems. Unfortunately, the level of abstraction of current mathematics often makes it difficult for anyone but an expert to appreciate this fact. Random Walks and Electric Networks looks at the interplay of physics and mathematics in terms of an example the relation between elementary electric network theory and random walks where the mathematics involved is at the college level.
1,632 citations
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01 Jan 1964
TL;DR: In this article, a very special class of random processes, namely to random walk on the lattice points of ordinary Euclidean space, is studied, and the author considered this high degree of specialization worth while because of the theory of such random walks is far more complete than that of any larger class of Markov chains.
Abstract: This book is devoted exclusively to a very special class of random processes, namely to random walk on the lattice points of ordinary Euclidean space. The author considered this high degree of specialization worth while, because of the theory of such random walks is far more complete than that of any larger class of Markov chains. The book will present no technical difficulties to the readers with some solid experience in analysis in two or three of the following areas: probability theory, real variables and measure, analytic functions, Fourier analysis, differential and integral operators. There are almost 100 pages of examples and problems.
1,605 citations
01 Jan 2001
TL;DR: Estimates on the important parameters of access time, commute time, cover time and mixing time are discussed and recent algorithmic applications of random walks are sketched, in particular to the problem of sampling.
Abstract: Various aspects of the theory of random walks on graphs are surveyed In particular, estimates on the important parameters of access time, commute time, cover time and mixing time are discussed Connections with the eigenvalues of graphs and with electrical networks, and the use of these connections in the study of random walks is described We also sketch recent algorithmic applications of random walks, in particular to the problem of sampling
1,564 citations
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TL;DR: The concept of quantum random walk is introduced, and it is shown that due to quantum interference effects the average path length can be much larger than the maximum allowed path in the corresponding classical random walk.
Abstract: We introduce the concept of quantum random walk, and show that due to quantum interference effects the average path length can be much larger than the maximum allowed path in the corresponding classical random walk A quantum-optics application is described
1,518 citations
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TL;DR: The mathematical theory behind the simple random walk is introduced and how this relates to Brownian motion and diffusive processes in general and a reinforced random walk can be used to model movement where the individual changes its environment.
Abstract: Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on the extensions of simple random walk processes. In this review paper, our aim is twofold: to introduce the mathematics behind random walks in a straightforward manner and to explain how such models can be used to aid our understanding of biological processes. We introduce the mathematical theory behind the simple random walk and explain how this relates to Brownian motion and diffusive processes in general. We demonstrate how these simple models can be extended to include drift and waiting times or be used to calculate first passage times. We discuss biased random walks and show how hyperbolic models can be used to generate correlated random walks. We cover two main applications of the random walk model. Firstly, we review models and results relating to the movement, dispersal and population redistribution of animals and micro-organisms. This includes direct calculation of mean squared displacement, mean dispersal distance, tortuosity measures, as well as possible limitations of these model approaches. Secondly, oriented movement and chemotaxis models are reviewed. General hyperbolic models based on the linear transport equation are introduced and we show how a reinforced random walk can be used to model movement where the individual changes its environment. We discuss the applications of these models in the context of cell migration leading to blood vessel growth (angiogenesis). Finally, we discuss how the various random walk models and approaches are related and the connections that underpin many of the key processes involved.
1,313 citations