Download or read book Aspects of Markov Chains and Particle Systems written by Shirshendu Ganguly and published by . This book was released on 2016 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: The thesis concerns asymptotic behavior of particle systems and the underlying Markov chains used to model various natural phenomena. The objective is to describe and analyze stochastic models involving spatial structure and evolution over time. Fundamental objects of interest in such systems include the equilibrium measure which the system converges to, the phenomenon of phase transition in the long term behavior and the time taken to converge to stationarity. In this thesis we present three examples highlighting the above aspects. In Chapter 2, we will discuss Competitive Erosion: a multi-particle system introduced by James Propp in 2003, as a generalization of a fundamental growth model known as Internal Diffusion Limited Aggregation. In this model, each vertex of the graph is occupied by a particle, which can be either red or blue. New red and blue particles are emitted alternately from their respective bases and perform random walk. On encountering a particle of the opposite color they remove it and occupy its position. We consider competitive erosion on discretizations of smooth planar simply connected domains. In Chapter 2 we establish positively, a conjecture of Propp regarding conformal invariance of the the model at stationarity, by showing that, with high probability the blue and the red regions are separated by an orthogonal circular arc on the disc and by a suitable hyperbolic geodesic on a general `smooth' simply connected domain. In Chapter 3, we discuss a family of conservative stochastic processes known as Activated Random Walk (ARW) which interpolates between ordinary random walk and the Stochastic Sandpile; the latter being a canonical example of Self Organized Criticality. These processes are conjectured to exhibit a sharp change in long time behavior depending on the value of certain parameters. Informally ARW is a particle system on Z with mass conservation. One starts with a mass density mu>0 of initially active particles, each of which performs a symmetric random walk at rate one and falls asleep at rate lambda. Sleepy particles become active on coming in contact with other active particles. We investigate the question of fixation/non-fixation of the process and show for small enough lambda, the critical mass density for fixation is strictly less than one. Moreover, the critical density goes to zero as lambda tends to zero. This positively answers two open questions from Dickman, Rolla, Sidoravicius (J. Stat. Phys., 2010) and Rolla, Sidoravicius (Invent. Math., 2012). In Chapter 4, we discuss a model of constrained Glauber dynamics, known as the East Process, exhibiting sharp convergence to equilibrium. The East process is a 1-D kinetically constrained interacting particle system, introduced in the physics literature in the early 90's to model liquid-glass transitions. Informally, it is a two spin (0,1) system on Z, where every site at rate one tries to randomize its spin using a fresh Bernoulli (p). However the move is suppressed unless the site to the left is in the 0 state. Thus the Glauber dynamics move is carried out only in the presence of a certain `kinetic' constraint. Spectral gap estimates of Aldous and Diaconis in 2002 imply that its mixing time on L sites has order L. Since the relaxation time is of a smaller order than the mixing time it is natural to expect a sharp convergence to equilibrium . Proving this, is the goal of this chapter, where we establish Cutoff for mixing, with an optimal window size.
Download or read book From Markov Chains to Non equilibrium Particle Systems written by Mufa Chen and published by World Scientific. This book was released on 2004 with total page 614 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is representative of the work of Chinese probabilists on probability theory and its applications in physics. It presents a unique treatment of general Markov jump processes: uniqueness, various types of ergodicity, Markovian couplings, reversibility, spectral gap, etc.
Download or read book From Markov Chains to Non equilibrium Particle Systems written by Mu Fa Chen and published by . This book was released on 1994 with total page 551 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book From Markov Chains to Non equilibrium Particle Systems written by Mu Fa Chen and published by . This book was released on 1992 with total page 550 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Interacting Particle Systems written by T.M. Liggett and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 499 pages. Available in PDF, EPUB and Kindle. Book excerpt: At what point in the development of a new field should a book be written about it? This question is seldom easy to answer. In the case of interacting particle systems, important progress continues to be made at a substantial pace. A number of problems which are nearly as old as the subject itself remain open, and new problem areas continue to arise and develop. Thus one might argue that the time is not yet ripe for a book on this subject. On the other hand, this field is now about fifteen years old. Many important of several basic models is problems have been solved and the analysis almost complete. The papers written on this subject number in the hundreds. It has become increasingly difficult for newcomers to master the proliferating literature, and for workers in allied areas to make effective use of it. Thus I have concluded that this is an appropriate time to pause and take stock of the progress made to date. It is my hope that this book will not only provide a useful account of much of this progress, but that it will also help stimulate the future vigorous development of this field.
Download or read book Genealogies Of Interacting Particle Systems written by Matthias Birkner and published by World Scientific. This book was released on 2020-02-24 with total page 363 pages. Available in PDF, EPUB and Kindle. Book excerpt: Interacting particle systems are Markov processes involving infinitely many interacting components. Since their introduction in the 1970s, researchers have found many applications in statistical physics and population biology. Genealogies, which follow the origin of the state of a site backwards in time, play an important role in their studies, especially for the biologically motivated systems.The program Genealogies of Interacting Particle Systems held at the Institute for Mathematical Sciences, National University of Singapore, from 17 July to 18 Aug 2017, brought together experts and young researchers interested in this modern topic. Central to the program were learning sessions where lecturers presented work outside of their own research, as well as a normal workshop. This is reflected in the present volume which contains two types of articles:Written by respected researchers, including experts in the field such as Steve Evans, member of the US National Academy of Sciences, as well as Anton Wakolbinger, Andreas Greven, and many others, this volume will no doubt be a valuable contribution to the probability community.
Download or read book On the Duality Relation Between Interacting Particle Systems and Markov Chains written by Anja Voß-Böhme and published by . This book was released on 2009 with total page 70 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book From Markov Chains to Nonequlibrium Particle Systems written by Mu Fa Chen and published by . This book was released on 1992 with total page 550 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Continuous Time Markov Processes written by Thomas Milton Liggett and published by American Mathematical Soc.. This book was released on 2010 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: Markov processes are among the most important stochastic processes for both theory and applications. This book develops the general theory of these processes, and applies this theory to various special examples.
Download or read book From Markov Chains to Non equilibrium Particle Systems written by Mufa Chen and published by World Scientific. This book was released on 2004 with total page 610 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is representative of the work of Chinese probabilists on probability theory and its applications in physics. It presents a unique treatment of general Markov jump processes: uniqueness, various types of ergodicity, Markovian couplings, reversibility, spectral gap, etc. It also deals with a typical class of non-equilibrium particle systems, including the typical Schlögl model taken from statistical physics. The constructions, ergodicity and phase transitions for this class of Markov interacting particle systems, namely, reaction-diffusion processes, are presented. In this new edition, a large part of the text has been updated and two-and-a-half chapters have been rewritten. The book is self-contained and can be used in a course on stochastic processes for graduate students.
Download or read book Additive and Cancellative Interacting Particle Systems written by D. Griffeath and published by Springer. This book was released on 2006-11-15 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt:
Download or read book Elements of the Theory of Markov Processes and Their Applications written by A. T. Bharucha-Reid and published by Courier Corporation. This book was released on 2012-04-26 with total page 485 pages. Available in PDF, EPUB and Kindle. Book excerpt: This graduate-level text and reference in probability, with numerous applications to several fields of science, presents nonmeasure-theoretic introduction to theory of Markov processes. The work also covers mathematical models based on the theory, employed in various applied fields. Prerequisites are a knowledge of elementary probability theory, mathematical statistics, and analysis. Appendixes. Bibliographies. 1960 edition.
Download or read book Discrete Time Markov Chains written by G. George Yin and published by Springer Science & Business Media. This book was released on 2005-10-04 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on two-time-scale Markov chains in discrete time. Our motivation stems from existing and emerging applications in optimization and control of complex systems in manufacturing, wireless communication, and ?nancial engineering. Much of our e?ort in this book is devoted to designing system models arising from various applications, analyzing them via analytic and probabilistic techniques, and developing feasible compu- tionalschemes. Ourmainconcernistoreducetheinherentsystemcompl- ity. Although each of the applications has its own distinct characteristics, all of them are closely related through the modeling of uncertainty due to jump or switching random processes. Oneofthesalientfeaturesofthisbookistheuseofmulti-timescalesin Markovprocessesandtheirapplications. Intuitively,notallpartsorcom- nents of a large-scale system evolve at the same rate. Some of them change rapidly and others vary slowly. The di?erent rates of variations allow us to reduce complexity via decomposition and aggregation. It would be ideal if we could divide a large system into its smallest irreducible subsystems completely separable from one another and treat each subsystem indep- dently. However, this is often infeasible in reality due to various physical constraints and other considerations. Thus, we have to deal with situations in which the systems are only nearly decomposable in the sense that there are weak links among the irreducible subsystems, which dictate the oc- sional regime changes of the system. An e?ective way to treat such near decomposability is time-scale separation. That is, we set up the systems as if there were two time scales, fast vs. slow. xii Preface Followingthetime-scaleseparation,weusesingularperturbationmeth- ology to treat the underlying systems.
Download or read book Phase Transitions Of Interacting Particle Systems written by Norio Konno and published by World Scientific. This book was released on 1995-01-16 with total page 245 pages. Available in PDF, EPUB and Kindle. Book excerpt: Recently, interacting particle systems have been studied widely from the standpoints of mathematics, physics, chemistry and biology. Many researchers are becoming interested in this field.This book focuses on the phase transitions of interacting particle systems, especially their critical values and order parameters. It poses the following question: How can we get good bounds on the critical values and the order parameters? This question is very basic, and many researchers have been trying to get better bounds rigorously. Hence the book provides bounds — both the author's and others'.
Download or read book Mixing Times of Interacting Particle Systems written by Rivka Maclaine Mitchell and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: "We study the asymptotic behavior of various Markov chains and their mixing times. For discrete-time Markov chains, we provide a proof of a version of Wormald's theorem, a result used to obtain asymptotics for a sequence of Markov chains. For continuous-time processes we study a specific class of Markov chains called interacting particle systems. We discuss recent work pertaining to the mixing times of interacting particle systems, and give a proof of a recent result by Jonathan Hermon and Justin Salez"--
Download or read book Random Walks Brownian Motion and Interacting Particle Systems written by H. Kesten and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 457 pages. Available in PDF, EPUB and Kindle. Book excerpt: This collection of articles is dedicated to Frank Spitzer on the occasion of his 65th birthday. The articles, written by a group of his friends, colleagues, former students and coauthors, are intended to demonstrate the major influence Frank has had on probability theory for the last 30 years and most likely will have for many years to come. Frank has always liked new phenomena, clean formulations and elegant proofs. He has created or opened up several research areas and it is not surprising that many people are still working out the consequences of his inventions. By way of introduction we have reprinted some of Frank's seminal articles so that the reader can easily see for himself the point of origin for much of the research presented here. These articles of Frank's deal with properties of Brownian motion, fluctuation theory and potential theory for random walks, and, of course, interacting particle systems. The last area was started by Frank as part of the general resurgence of treating problems of statistical mechanics with rigorous probabilistic tools.
Download or read book Continuous Time Markov Processes written by Thomas Milton Liggett and published by American Mathematical Soc.. This book was released on 2010-01-01 with total page 288 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Markov processes are among the most important stochastic processes for both theory and applications. This book develops the general theory of these processes and applies this theory to various special examples. The initial chapter is devoted to the most important classical example--one-dimensional Brownian motion. This, together with a chapter on continuous time Markov chains, provides the motivation for the general setup based on semigroups and generators. Chapters on stochastic calculus and probabilistic potential theory give an introduction to some of the key areas of application of Brownian motion and its relatives. A chapter on interacting particle systems treats a more recently developed class of Markov processes that have as their origin problems in physics and biology."--Publisher's description.
