Random Walks on Infinite Graphs and Groups
Random Walks on Infinite Graphs and Groups

Author: Wolfgang Woess

Publisher: Cambridge University Press

Published: 2000-02-13

Total Pages: 350

ISBN-13: 0521552923

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The main theme of this book is the interplay between the behaviour of a class of stochastic processes (random walks) and discrete structure theory. The author considers Markov chains whose state space is equipped with the structure of an infinite, locally finite graph, or as a particular case, of a finitely generated group. The transition probabilities are assumed to be adapted to the underlying structure in some way that must be specified precisely in each case. From the probabilistic viewpoint, the question is what impact the particular type of structure has on various aspects of the behaviour of the random walk. Vice-versa, random walks may also be seen as useful tools for classifying, or at least describing the structure of graphs and groups. Links with spectral theory and discrete potential theory are also discussed. This book will be essential reading for all researchers working in stochastic process and related topics.

Random Walks on Infinite Graphs and Groups
Random Walks on Infinite Graphs and Groups

Author: Wolfgang Woess

Publisher: Cambridge University Press

Published: 2008-05-19

Total Pages: 0

ISBN-13: 9780521061728

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This eminent work focuses on the interplay between the behavior of random walks and discrete structure theory. Wolfgang Woess considers Markov chains whose state space is equipped with the structure of an infinite, locally-finite graph, or of a finitely generated group. He assumes the transition probabilities are adapted to the underlying structure in some way that must be specified precisely in each case. He also explores the impact the particular type of structure has on various aspects of the behavior of the random walk. In addition, the author shows how random walks are useful tools for classifying, or at least describing, the structure of graphs and groups.

Random Walks on Infinite Groups
Random Walks on Infinite Groups

Author: Steven P. Lalley

Publisher: Springer Nature

Published: 2023-05-08

Total Pages: 373

ISBN-13: 3031256328

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This text presents the basic theory of random walks on infinite, finitely generated groups, along with certain background material in measure-theoretic probability. The main objective is to show how structural features of a group, such as amenability/nonamenability, affect qualitative aspects of symmetric random walks on the group, such as transience/recurrence, speed, entropy, and existence or nonexistence of nonconstant, bounded harmonic functions. The book will be suitable as a textbook for beginning graduate-level courses or independent study by graduate students and advanced undergraduate students in mathematics with a solid grounding in measure theory and a basic familiarity with the elements of group theory. The first seven chapters could also be used as the basis for a short course covering the main results regarding transience/recurrence, decay of return probabilities, and speed. The book has been organized and written so as to be accessible not only to students in probability theory, but also to students whose primary interests are in geometry, ergodic theory, or geometric group theory.

Probability on Trees and Networks
Probability on Trees and Networks

Author: Russell Lyons

Publisher: Cambridge University Press

Published: 2017-01-20

Total Pages: 1023

ISBN-13: 1316785335

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Starting around the late 1950s, several research communities began relating the geometry of graphs to stochastic processes on these graphs. This book, twenty years in the making, ties together research in the field, encompassing work on percolation, isoperimetric inequalities, eigenvalues, transition probabilities, and random walks. Written by two leading researchers, the text emphasizes intuition, while giving complete proofs and more than 850 exercises. Many recent developments, in which the authors have played a leading role, are discussed, including percolation on trees and Cayley graphs, uniform spanning forests, the mass-transport technique, and connections on random walks on graphs to embedding in Hilbert space. This state-of-the-art account of probability on networks will be indispensable for graduate students and researchers alike.

Probability on Graphs
Probability on Graphs

Author: Geoffrey Grimmett

Publisher: Cambridge University Press

Published: 2018-01-25

Total Pages: 279

ISBN-13: 1108542999

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This introduction to some of the principal models in the theory of disordered systems leads the reader through the basics, to the very edge of contemporary research, with the minimum of technical fuss. Topics covered include random walk, percolation, self-avoiding walk, interacting particle systems, uniform spanning tree, random graphs, as well as the Ising, Potts, and random-cluster models for ferromagnetism, and the Lorentz model for motion in a random medium. This new edition features accounts of major recent progress, including the exact value of the connective constant of the hexagonal lattice, and the critical point of the random-cluster model on the square lattice. The choice of topics is strongly motivated by modern applications, and focuses on areas that merit further research. Accessible to a wide audience of mathematicians and physicists, this book can be used as a graduate course text. Each chapter ends with a range of exercises.

Random Graph Dynamics
Random Graph Dynamics

Author: Rick Durrett

Publisher: Cambridge University Press

Published: 2010-05-31

Total Pages: 203

ISBN-13: 1139460889

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The theory of random graphs began in the late 1950s in several papers by Erdos and Renyi. In the late twentieth century, the notion of six degrees of separation, meaning that any two people on the planet can be connected by a short chain of people who know each other, inspired Strogatz and Watts to define the small world random graph in which each site is connected to k close neighbors, but also has long-range connections. At a similar time, it was observed in human social and sexual networks and on the Internet that the number of neighbors of an individual or computer has a power law distribution. This inspired Barabasi and Albert to define the preferential attachment model, which has these properties. These two papers have led to an explosion of research. The purpose of this book is to use a wide variety of mathematical argument to obtain insights into the properties of these graphs. A unique feature is the interest in the dynamics of process taking place on the graph in addition to their geometric properties, such as connectedness and diameter.

Random Walks and Geometry
Random Walks and Geometry

Author: Vadim Kaimanovich

Publisher: Walter de Gruyter

Published: 2008-08-22

Total Pages: 545

ISBN-13: 3110198088

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Die jüngsten Entwicklungen zeigen, dass sich Wahrscheinlichkeitsverfahren zu einem sehr wirkungsvollen Werkzeug entwickelt haben, und das auf so unterschiedlichen Gebieten wie statistische Physik, dynamische Systeme, Riemann'sche Geometrie, Gruppentheorie, harmonische Analyse, Graphentheorie und Informatik.

Random Graphs and Complex Networks
Random Graphs and Complex Networks

Author: Remco van der Hofstad

Publisher: Cambridge University Press

Published: 2017

Total Pages: 341

ISBN-13: 110717287X

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This classroom-tested text is the definitive introduction to the mathematics of network science, featuring examples and numerous exercises.

Random Walks and Electric Networks
Random Walks and Electric Networks

Author: Peter G. Doyle

Publisher: American Mathematical Soc.

Published: 1984-12-31

Total Pages: 159

ISBN-13: 1614440220

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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.