Symmetric Random Walks On Groups

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Symmetric Random Walks On Groups

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Author : Harry Kesten
language : en
Publisher:
Release Date : 1958

Symmetric Random Walks On Groups written by Harry Kesten and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1958 with Group theory categories.

Random Walks On Infinite Groups

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Author : Steven P. Lalley
language : en
Publisher: Springer Nature
Release Date : 2023-05-08

Random Walks On Infinite Groups written by Steven P. Lalley and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2023-05-08 with Mathematics categories.

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.

Random Walks On The Symmetric Group Likelihood Orders And Involutions

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Author : Megan Marla Bernstein
language : en
Publisher:
Release Date : 2015

Random Walks On The Symmetric Group Likelihood Orders And Involutions written by Megan Marla Bernstein and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015 with categories.

This thesis delves into a largely unexplored concept in Markov chain analysis called a likelihood order. A likelihood order for time t of a Markov chain, as defined in this thesis, is an extension of the partial order from most likely to least likely at time t on the states to a total order. Random walks on finite groups are a subset of Markov chains. Aperiodic, irreducible random walks on finite groups converge to uniform over the group. The likelihood order captures internal structure of the walk that is resisting mixing. Understanding the most likely element, least likely element, and other parts of the likelihood order can aid in calculating the total variation distance, separation distance, and l infinity distance of the walk. After sufficient time, the likelihood order for time t will often converge to a fixed likelihood order. For random walks on groups the likelihood order for time t can be proved in some cases with induction and for conjugacy class walks using the discrete Fourier transform and the representation theory of the group. The thesis starts by examining the likelihood orders for simple random walk on cycles, products of cycles, and the dihedral group using both techniques whenever possible. The likelihood orders after sufficient time are identified for a number of random walks on the symmetric group. These walks can all be realized as shuffles of n cards on a table. The likelihood orders that appear are all variants of the cycle lexicographic order on the conjugacy classes of the symmetric group. The transposition walk on the symmetric group is proven to have likelihood order the cycle lexicographic order after order n squared steps. If the walk is made lazy, the likelihood order is shown to vary considerably based on the degree of laziness. The 3-cycle walk on the symmetric group is shown to have an alternating cycle lexicographic order after order n cubed steps. The n-cycle walk for time order n log(n) has an alternating cycle lexicographic order at even times and the reverse of the cycle lexicographic order at odd times. The most and least likely elements of the walk are fixed by 8 steps as the first and last elements of these orders. For p greater than or equal to 4 fixed, n sufficiently large, the likelihood order after sufficient time is found to be a piecewise combination of the cycle lexicographic order and alternating cycle lexicographic order. The likelihood order after sufficient time is also found for the following much more complicated walk. Consider n cards on a table. Pick a random perfect matching of the cards into pairs. Ignore each pair with probability p. For each remaining pair, transpose the cards positions on the table. This makes one step of the involution walk. This walk also has the cycle lexicographic order as its likelihood order after sufficient time. The involution walk is generated by involutions of the symmetric group with binomially distributed 2-cycles. It was introduced to study a conjecture about a random walk on the unitary group from the information theory of back holes. The question of interest is if you take a specific random walk with mixing time n log(n), and then modify it by at each time choosing n/2 commuting generators, how does the mixing time change. The involution walk is shown in this thesis to mix for p at least one half fixed, n sufficiently large in between log based 1/p of n steps and log base 2/(1+p) of n steps. This is a toy model for the unitary group problem, since it selects n/2 commuting generators from the half lazy transposition walk. The observed speed up is O(n) as predicted. The thesis introduces a new technique for finding eigenvalues of random walks generated by many conjugacy classes using the character polynomial for the characters of the representations of the symmetric group. This is especially efficient at calculating the large eigenvalues. The smaller eigenvalues are handled by developing monotonicity relations that also give the likelihood order after sufficient time for the walk.

Random Walks On Solvable Groups

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Author : David Robert Revelle
language : en
Publisher:
Release Date : 2002

Random Walks On Solvable Groups written by David Robert Revelle and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2002 with categories.

Random Walks On The Symmetric Group

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Author : Oliver Matheau-Raven
language : en
Publisher:
Release Date : 2020

Random Walks On The Symmetric Group written by Oliver Matheau-Raven and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2020 with categories.

Random Walks And Discrete Potential Theory

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Author : M. Picardello
language : en
Publisher: Cambridge University Press
Release Date : 1999-11-18

Random Walks And Discrete Potential Theory written by M. Picardello and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 1999-11-18 with Mathematics categories.

Comprehensive and interdisciplinary text covering the interplay between random walks and structure theory.

Random Walks On Infinite Graphs And Groups

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Author : Wolfgang Woess
language : en
Publisher: Cambridge University Press
Release Date : 2000-02-13

Random Walks On Infinite Graphs And Groups written by Wolfgang Woess and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2000-02-13 with Mathematics categories.

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.

Harmonic Functions And Random Walks On Groups

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Author : Ariel Yadin
language : en
Publisher: Cambridge University Press
Release Date : 2024-05-31

Harmonic Functions And Random Walks On Groups written by Ariel Yadin and has been published by Cambridge University Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2024-05-31 with Mathematics categories.

A modern introduction into the emerging research field of harmonic functions and random walks on groups.

Random Walks On The Symmetric Group Generated By Conjugacy Classes

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Author : Nathan Anton Mikerin Lulov
language : en
Publisher:
Release Date : 1996

Random Walks On The Symmetric Group Generated By Conjugacy Classes written by Nathan Anton Mikerin Lulov and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1996 with Fourier transformations categories.

Random Walk In Random And Non Random Environments Third Edition

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Author : Pal Revesz
language : en
Publisher: World Scientific
Release Date : 2013-03-06

Random Walk In Random And Non Random Environments Third Edition written by Pal Revesz and has been published by World Scientific this book supported file pdf, txt, epub, kindle and other format this book has been release on 2013-03-06 with Mathematics categories.

The simplest mathematical model of the Brownian motion of physics is the simple, symmetric random walk. This book collects and compares current results — mostly strong theorems which describe the properties of a random walk. The modern problems of the limit theorems of probability theory are treated in the simple case of coin tossing. Taking advantage of this simplicity, the reader is familiarized with limit theorems (especially strong ones) without the burden of technical tools and difficulties. An easy way of considering the Wiener process is also given, through the study of the random walk.Since the first and second editions were published in 1990 and 2005, a number of new results have appeared in the literature. The first two editions contained many unsolved problems and conjectures which have since been settled; this third, revised and enlarged edition includes those new results. In this edition, a completely new part is included concerning Simple Random Walks on Graphs. Properties of random walks on several concrete graphs have been studied in the last decade. Some of the obtained results are also presented.