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• Simple Random Walk
• Harmonic Measure
• Intersection Probabilities
• Four Dimensions
• Two and Three Dimensions
• Self-Avoiding Walks
• Loop-Erased Walk.

A central study in Probability Theory is the behavior of fluctuation phenomena of partial sums of different types of random variable. One of the most useful concepts for this purpose is that of the random walk which has applications in many areas, particularly in statistical physics and statistical chemistry. Originally published in 1991, Intersections of Random Walks focuses on and explores a number of problems dealing primarily with the nonintersection of random walks and the self-avoiding walk. Many of these problems arise in studying statistical physics and other critical phenomena. Topics include: discrete harmonic measure, including an introduction to diffusion limited aggregation (DLA); the probability that independent random walks do not intersect; and properties of walks without self-intersections. The present softcover reprint includes corrections and addenda from the 1996 printing, and makes this classic monograph available to a wider audience. With a self-contained introduction to the properties of simple random walks, and an emphasis on rigorous results, the book will be useful to researchers in probability and statistical physics and to graduate students interested in basic properties of random walks.
(source: Nielsen Book Data)

• Preface--
• 1. Introduction--
• 2. Local central limit theorem--
• 3. Approximation by Brownian motion--
• 4. Green's function--
• 5. One-dimensional walks--
• 6. Potential theory--
• 7. Dyadic coupling--
• 8. Additional topics on simple random walk--
• 9. Loop measures--
• 10. Intersection probabilities for random walks--
• 11. Loop-erased random walk-- Appendix-- Bibliography-- Index of symbols-- Index.
• (source: Nielsen Book Data)

Random walks are stochastic processes formed by successive summation of independent, identically distributed random variables and are one of the most studied topics in probability theory. This contemporary introduction evolved from courses taught at Cornell University and the University of Chicago by the first author, who is one of the most highly regarded researchers in the field of stochastic processes. This text meets the need for a modern reference to the detailed properties of an important class of random walks on the integer lattice. It is suitable for probabilists, mathematicians working in related fields, and for researchers in other disciplines who use random walks in modeling.
(source: Nielsen Book Data)

• Preface--
• 1. Introduction--
• 2. Local central limit theorem--
• 3. Approximation by Brownian motion--
• 4. Green's function--
• 5. One-dimensional walks--
• 6. Potential theory--
• 7. Dyadic coupling--
• 8. Additional topics on simple random walk--
• 9. Loop measures--
• 10. Intersection probabilities for random walks--
• 11. Loop-erased random walk-- Appendix-- Bibliography-- Index of symbols-- Index.
• (source: Nielsen Book Data)

Random walks are stochastic processes formed by successive summation of independent, identically distributed random variables and are one of the most studied topics in probability theory. This contemporary introduction evolved from courses taught at Cornell University and the University of Chicago by the first author, who is one of the most highly regarded researchers in the field of stochastic processes. This text meets the need for a modern reference to the detailed properties of an important class of random walks on the integer lattice. It is suitable for probabilists, mathematicians working in related fields, and for researchers in other disciplines who use random walks in modeling.
(source: Nielsen Book Data)

• Preface to Second Edition Preface to First Edition PRELIMINARIES Introduction Linear Differential Equations Linear Difference Equations Exercises FINITE MARKOV CHAINS Definitions and Examples Large-Time Behavior and Invariant Probability Classification of States Return Times Transient States Examples Exercises COUNTABLE MARKOV CHAINS Introduction Recurrence and Transience Positive Recurrence and Null Recurrence Branching Process Exercises CONTINUOUS-TIME MARKOV CHAINS Poisson Process Finite State Space Birth-and-Death Processes General Case Exercises OPTIMAL STOPPING Optimal Stopping of Markov Chains Optimal Stopping with Cost Optimal Stopping with Discounting Exercises MARTINGALES Conditional Expectation Definition and Examples Optional Sampling Theorem Uniform Integrability Martingale Convergence Theorem Maximal Inequalities Exercises RENEWAL PROCESSES Introduction Renewal Equation Discrete Renewal Processes M/G/1 and G/M/1 Queues Exercises REVERSIBLE MARKOV CHAINS Reversible Processes Convergence to Equilibrium Markov Chain Algorithms A Criterion for Recurrence Exercises BROWNIAN MOTION Introduction Markov Property Zero Set of Brownian Motion Brownian Motion in Several Dimensions Recurrence and Transience Fractal Nature of Brownian Motion Scaling Rules Brownian Motion with Drift Exercises STOCHASTIC INTEGRATION Integration with Respect to Random Walk Integration with Respect to Brownian Motion Ito's Formula Extensions if Ito's Formula Continuous Martingales Girsanov Transformation Feynman-Kac Formula Black-Scholes Formula Simulation Exercises Suggestions for Further Reading Index.
• (source: Nielsen Book Data)

Emphasizing fundamental mathematical ideas rather than proofs, Introduction to Stochastic Processes, Second Edition provides quick access to important foundations of probability theory applicable to problems in many fields. Assuming that you have a reasonable level of computer literacy, the ability to write simple programs, and the access to software for linear algebra computations, the author approaches the problems and theorems with a focus on stochastic processes evolving with time, rather than a particular emphasis on measure theory. For those lacking in exposure to linear differential and difference equations, the author begins with a brief introduction to these concepts. He proceeds to discuss Markov chains, optimal stopping, martingales, and Brownian motion. The book concludes with a chapter on stochastic integration. The author supplies many basic, general examples and provides exercises at the end of each chapter. New to the Second Edition: Expanded chapter on stochastic integration that introduces modern mathematical finance Introduction of Girsanov transformation and the Feynman-Kac formula Expanded discussion of Ito's formula and the Black-Scholes formula for pricing options New topics such as Doob's maximal inequality and a discussion on self similarity in the chapter on Brownian motion Applicable to the fields of mathematics, statistics, and engineering as well as computer science, economics, business, biological science, psychology, and engineering, this concise introduction is an excellent resource both for students and professionals.
(source: Nielsen Book Data)

• Preface to Second Edition Preface to First Edition PRELIMINARIES Introduction Linear Differential Equations Linear Difference Equations Exercises FINITE MARKOV CHAINS Definitions and Examples Large-Time Behavior and Invariant Probability Classification of States Return Times Transient States Examples Exercises COUNTABLE MARKOV CHAINS Introduction Recurrence and Transience Positive Recurrence and Null Recurrence Branching Process Exercises CONTINUOUS-TIME MARKOV CHAINS Poisson Process Finite State Space Birth-and-Death Processes General Case Exercises OPTIMAL STOPPING Optimal Stopping of Markov Chains Optimal Stopping with Cost Optimal Stopping with Discounting Exercises MARTINGALES Conditional Expectation Definition and Examples Optional Sampling Theorem Uniform Integrability Martingale Convergence Theorem Maximal Inequalities Exercises RENEWAL PROCESSES Introduction Renewal Equation Discrete Renewal Processes M/G/1 and G/M/1 Queues Exercises REVERSIBLE MARKOV CHAINS Reversible Processes Convergence to Equilibrium Markov Chain Algorithms A Criterion for Recurrence Exercises BROWNIAN MOTION Introduction Markov Property Zero Set of Brownian Motion Brownian Motion in Several Dimensions Recurrence and Transience Fractal Nature of Brownian Motion Scaling Rules Brownian Motion with Drift Exercises STOCHASTIC INTEGRATION Integration with Respect to Random Walk Integration with Respect to Brownian Motion Ito's Formula Extensions if Ito's Formula Continuous Martingales Girsanov Transformation Feynman-Kac Formula Black-Scholes Formula Simulation Exercises Suggestions for Further Reading Index.
• (source: Nielsen Book Data)

Emphasizing fundamental mathematical ideas rather than proofs, Introduction to Stochastic Processes, Second Edition provides quick access to important foundations of probability theory applicable to problems in many fields. Assuming that you have a reasonable level of computer literacy, the ability to write simple programs, and the access to software for linear algebra computations, the author approaches the problems and theorems with a focus on stochastic processes evolving with time, rather than a particular emphasis on measure theory. For those lacking in exposure to linear differential and difference equations, the author begins with a brief introduction to these concepts. He proceeds to discuss Markov chains, optimal stopping, martingales, and Brownian motion. The book concludes with a chapter on stochastic integration. The author supplies many basic, general examples and provides exercises at the end of each chapter. New to the Second Edition: Expanded chapter on stochastic integration that introduces modern mathematical finance Introduction of Girsanov transformation and the Feynman-Kac formula Expanded discussion of Ito's formula and the Black-Scholes formula for pricing options New topics such as Doob's maximal inequality and a discussion on self similarity in the chapter on Brownian motion Applicable to the fields of mathematics, statistics, and engineering as well as computer science, economics, business, biological science, psychology, and engineering, this concise introduction is an excellent resource both for students and professionals.
(source: Nielsen Book Data)

• Some discrete processes Stochastic calculus Complex Brownian motion Conformal mappings Loewner differential equation Brownian measures on paths Schramm-Loewner evolution More results about SLE Brownian intersection exponent Restriction measures Hausdorff dimension Hypergeometric functions Reflecting Brownian motion Bibliography Index Index of symbols.
• (source: Nielsen Book Data)

Theoretical physicists have predicted that the scaling limits of many two-dimensional lattice models in statistical physics are in some sense conformally invariant. This belief has allowed physicists to predict many quantities for these critical systems. The nature of these scaling limits has recently been described precisely by using one well-known tool, Brownian motion, and a new construction, the Schramm-Loewner evolution (SLE). This book is an introduction to the conformally invariant processes that appear as scaling limits. The following topics are covered: stochastic integration; complex Brownian motion and measures derived from Brownian motion; conformal mappings and univalent functions; the Loewner differential equation and Loewner chains; the Schramm-Loewner evolution (SLE), which is a Loewner chain with a Brownian motion input; and, applications to intersection exponents for Brownian motion.The prerequisites are first-year graduate courses in real analysis, complex analysis, and probability. The book is suitable for graduate students and research mathematicians interested in random processes and their applications in theoretical physics.
(source: Nielsen Book Data)

• 1. Stochastic calculus 2. Complex Brownian motion 3. Conformal mappings 4. Loewner differential equation 5. Brownian measures on paths 6. Schramm-Loewner evolution 7. More results about $SLE$ 8. Brownian intersection exponent 9. Restriction measures

Theoretical physicists have predicted that the scaling limits of many two-dimensional lattice models in statistical physics are in some sense conformally invariant. This belief has allowed physicists to predict many quantities for these critical systems. The nature of these scaling limits has recently been described precisely by using one well-known tool, Brownian motion, and a new construction, the Schramm-Loewner evolution (SLE). This book is an introduction to the conformally invariant processes that appear as scaling limits. The following topics are covered: stochastic integration; complex Brownian motion and measures derived from Brownian motion; conformal mappings and univalent functions; the Loewner differential equation and Loewner chains; the Schramm-Loewner evolution (SLE), which is a Loewner chain with a Brownian motion input; and applications to intersection exponents for Brownian motion. The prerequisites are first-year graduate courses in real analysis, complex analysis, and probability. The book is suitable for graduate students and research mathematicians interested in random processes and their applications in theoretical physics.
(source: Nielsen Book Data)

• Simple random walk and Stirling's formula Simple ramdon walk in many dimensions Self-avoiding walk Brownian motion Shuffling and random permutations Seven shuffles are enough (sort of) Markov chains on finite sets Markov chain Monte Carlo Random walks and electrical networks Uniform spanning trees Random walk simulations Other simulations Simulations in finance Problems Bibliography.
• (source: Nielsen Book Data)

This volume is based on classes in probability for advanced undergraduates held at the IAS/Park City Mathematics Institute (Utah). It is derived from both lectures (Chapters 1-10) and computer simulations (Chapters 11-13) that were held during the program. The material is coordinated so that some of the major computer simulations relate to topics covered in the first ten chapters. The goal is to present topics that are accessible to advanced undergraduates, yet are areas of current research in probability. The combination of the lucid yet informal style of the lectures and the hands-on nature of the simulations allows readers to become familiar with some interesting and active areas of probability. The first four chapters discuss random walks and the continuous limit of random walks: Brownian motion.Chapters 5 and 6 consider the fascinating mathematics of card shuffles, including the notions of random walks on a symmetric group and the general idea of random permutations. Chapters 7 and 8 discuss Markov chains, beginning with a standard introduction to the theory. Chapter 8 addresses the recent important application of Markov chains to simulations of random systems on large finite sets: Markov Chain Monte Carlo. Random walks and electrical networks are covered in Chapter 9. Uniform spanning trees, as connected to probability and random walks, are treated in Chapter 10. The final three chapters of the book present simulations. Chapter 11 discusses simulations for random walks.Chapter 12 covers simulation topics such as sampling from continuous distributions, random permutations, and estimating the number of matrices with certain conditions using Markov Chain Monte Carlo. Chapter 13 presents simulations of stochastic differential equations for applications in finance. (The simulations do not require one particular piece of software. They can be done in symbolic computation packages or via programming languages such as C.) The volume concludes with a number of problems ranging from routine to very difficult. Of particular note are problems that are typical of simulation problems given to students by the authors when teaching undergraduate probability.
(source: Nielsen Book Data)

Focusing on a number of problems related to the intersection of random walks and the self-avoiding walk. Many of these problems arise in studying statistical physics and other critical phenomena. Topics covered include: discrete harmonic measure; the probability that independent random walks do not intersect; and properties of walks without self-intersections. With the inclusion of a self-contained introduction to the properties of simple random walks, and an emphasis on rigorous results, this text should be of use to researchers in probability and statistical physics, and to graduate students interested in basic properties of random walk.
(source: Nielsen Book Data)

• PRELIMINARIESIntroductionLinear Differential EquationsLinear Difference EquationsFINITE MARKOV CHAINSDefinitions and ExamplesLong-Range Behavior and Invariant ProbabilityClassification of StatesReturn TimesTransient StatesExamplesCOUNTABLE MARKOV CHAINSIntroductionRecurrence and TransiencePositive Recurrence and Null RecurrenceBranching ProcessCONTINUOUS-TIME MARKOV CHAINSPoisson ProcessFinite State SpaceBirth-and-Death ProcessesGeneral CaseOPTIMAL STOPPINGOptimal Stopping of Markov ChainsOptimal Stopping with CostOptimal Stopping with DiscountingMARTINGALESConditional ExpectationDefinition and ExamplesOptional Sampling theoremUniform IntegrabilityMartingale Convergence TheoremRENEWAL PROCESSESIntroductionRenewal EquationDiscrete Renewal ProcessesM/G/1 and B/M/1 QueuesREVERSIBLE MARKOV CHAINSReversible ProcessesConvergence to EquilibriumMarkov Chain AlgorithmsA Criterion for RecurrenceBROWNIAN MOTIONIntroductionMarkov PropertyZero Set of Brownian MotionBrownian Motion in Several DimensionsRecurrence and TransienceFractal Nature of Brownian MotionBrownian Motion with DriftSTOCHASTIC INTEGRATIONIntegration with Respect to Random walkIntegration with Respect to Brownian MotionIto's FormulaSimulationINDEX.
• (source: Nielsen Book Data)

This concise, informal introduction to stochastic processes evolving with time was designed to meet the needs of graduate students not only in mathematics and statistics, but in the many fields in which the concepts presented are important, including computer science, economics, business, biological science, psychology, and engineering. With emphasis on fundamental mathematical ideas rather than proofs or detailed applications, the treatment introduces the following topics: Markov chains, with focus on the relationship between the convergence to equilibrium and the size of the eigenvalues of the stochastic matrix Infinite state space, including the ideas of transience, null recurrence and positive recurrence The three main types of continual time Markov chains and optimal stopping of Markov chains Martingales, including conditional expectation, the optional sampling theorem, and the martingale convergence theorem Renewal process and reversible Markov chains Brownian motion, both multidimensional and one-dimensionalIntroduction to Stochastic Processes is ideal for a first course in stochastic processes without measure theory, requiring only a calculus-based undergraduate probability course and a course in linear algebra.
(source: Nielsen Book Data)

• Anton Bovier: From spin glasses to branching Brownian motion - and back?.- David Brydges: The Renormalization Group and Self-AvoidingWalk.- Amin Coja-Oghlan: Phase Transitions in Discrete Structures.- Dmitry Ioffe: Multidimensional Random Polymers : A Renewal Approach.- Gregory F. Lawler and Jacob Perlman: Loop measures and the Gaussian free field.
• (source: Nielsen Book Data)

Focusing on the mathematics that lies at the intersection of probability theory, statistical physics, combinatorics and computer science, this volume collects together lecture notes on recent developments in the area. The common ground of these subjects is perhaps best described by the three terms in the title: Random Walks, Random Fields and Disordered Systems. The specific topics covered include a study of Branching Brownian Motion from the perspective of disordered (spin-glass) systems, a detailed analysis of weakly self-avoiding random walks in four spatial dimensions via methods of field theory and the renormalization group, a study of phase transitions in disordered discrete structures using a rigorous version of the cavity method, a survey of recent work on interacting polymers in the ballisticity regime and, finally, a treatise on two-dimensional loop-soup models and their connection to conformally invariant systems and the Gaussian Free Field. The notes are aimed at early graduate students with a modest background in probability and mathematical physics, although they could also be enjoyed by seasoned researchers interested in learning about recent advances in the above fields.
(source: Nielsen Book Data)