1.1. From the Publisher of the 2013 reprint.

A central study in Probability Theory is the behaviour 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 non-intersection 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.

1.2. Extract from the Preface.

A more accurate title for this book would be "Problems dealing with the non-intersection of paths of random walks." These include: harmonic measure, which can be considered as a problem of non-intersection of a random walk with a fixed set; the probability that the paths of independent random walks do not intersect; and self-avoiding walks, i.e., random walks which have no self-intersections. The prerequisite is a standard measure theoretic course in probability including martingales and Brownian motion.

The first chapter develops the facts about simple random walk that will be needed. The discussion is self-contained although some previous exposure to random walks would be helpful. Many of the results are standard, and I have borrowed from a number of sources, especially the excellent book of Spitzer. For the sake of simplicity I have restricted the discussion to simple random walk. Of course, many of the results hold equally well for more general walks. ...

The proof of the local central limit theorem in Section 1.2 follows closely the proof in Spitzer. The next sections develop the usual probabilistic tools for analysing walks: stopping times, the strong Markov property, martingales derived from random walks, and boundary value problems for discrete harmonic functions. Again, all of this material is standard. ...

Harmonic measure is the subject of the second chapter. By harmonic measure here we mean harmonic measure from infinity, i.e., the hitting distribution of a set from a random walker starting at infinity. There are many ways to show the existence of harmonic measure, see e.g. Spitzer. Here the existence is derived as a consequence of the results in Section 1.7. This method has the advantage that it gives a bound on the rate of convergence. In Sections 2.2 and 2.3, the idea of discrete capacity is developed. The results of these sections are well known although some of the proofs are new. I take the viewpoint here that capacity is a measure of the probability that a random walk will hit a set. In the process, I completely ignore the interpretation in terms of electrical capacity or equilibrium potentials. Computing harmonic measure or escape probabilities can be very difficult. ...

The next three chapters study the problem of intersections of random walks or, more precisely, the probability that the paths of independent random walks intersect. We will not discuss in detail what the typical intersection set looks like. This has been studied by a number of authors under the name "intersection local time"...

The techniques of Chapter 3 are not powerful enough to analyse the probability that two one-sided walks starting at the origin do not intersect. There are a number of reasons to be interested in this problem. It is a random walk analogue of a quantity that arises in a number of problems in mathematical physics (e.g., a similar quantity arises in the discussion of critical exponents for self-avoiding walks in Section 6.3). Also, some of the techniques used in non-rigorous calculations in mathematical physics can be applied to this problem, so rigorous analysis of this problem can be used as a test of the effectiveness of these non-rigorous methods. ...

In four dimensions, the probability of non-intersection goes to zero like an inverse power of the logarithm of the number of steps. The techniques of Chapter 3 give bounds on this power; in Chapter 4, the exact power is derived. The first part of the derivation is to give asymptotic expressions for the probability of "long-range" intersections (the results of the previous chapter only give expressions up to a multiplicative constant). ...

The next chapter considers the intersection probability in dimensions two and three. Here the probability of no intersection goes to zero like a power of the number of steps. Again, the results of Chapter 3 can be used to give upper and lower bounds for the exponent. The first thing that is proved is that the exponent exists. ...

The last two chapters are devoted to self-avoiding walks, i.e., random walks conditioned to have no (or few) self-intersections. Sections 6.2 and 6.3 discuss the usual (strictly) self-avoiding walk, i.e., simple random walk of a given length with no self-intersections. The connective constant is defined, and then there is a discussion of the critical exponents for the model. The critical exponents are discussed from a probabilistic viewpoint; however, the discussion is almost entirely heuristic. ...

The last chapter discusses a particular model for self-avoiding walks, the loop-erased or Laplacian random walk. This model can be defined in two equivalent ways, one by erasing loops from the paths of simple random walk and the other as a kinetically growing walks with steps taken weighted according to harmonic measure. This model is similar to the usual self-avoiding walk in a number of ways: the critical dimension is four; there is convergence to Brownian motion for dimensions greater than or equal to four, with a logarithmic correction in four dimensions; nontrivial exponents describe the mean-square displacement below four dimensions. Unfortunately, this walk is not in the same universality class as the usual self-avoiding walk; in particular, the mean-square displacement exponent is different. The basic construction of the process is done in the first four sections. There are some technical difficulties in defining the walk in two dimensions because of the recurrence of simple random walk. ...

1.3. Review by: D Dhar.
Current Science 73 (5) (1997), 474.

The book by Lawler, as noted in his preface, deals more precisely with 'problems dealing with non-intersection of paths of random walks'. One of the questions which turns out to be important in the theory of random walks is whether there is a nonzero probability that a random walker will never return to the starting point again. It was shown by Polya that in one and two dimensions, this probability is exactly zero. But if the walker is bound to return to the origin at least once, he is bound to return again, and again. This, in turn, implies that in one and two dimensions, a random walker 'eventually' visits all sites infinitely often. In higher dimensions, there is a finite probability that a walker will not visit any particular prechosen site at any later time, however long we wait. Thus the behaviour of random walker in d dimensions is quite different for $d ≤ 2$ and $d > 2$.

Polya's theorem concerns the probability of intersection of the path of a random walker with a fixed point in space (say the origin). One can extend the treatment to the case when the walker is instructed to stop as soon as it hits anyone of a prespecified set of points. What is the distribution of probabilities of different stopping points? Such probability measures, when the initial point of walker is very far from the region where stopping points are, are called harmonic measures. These often are important in physics, e.g. in the study of diffusion-limited aggregates, which models the aggregation of soot particles in air. To characterise the average properties of these diffusion-limited aggregates (such as the fractal dimension), one needs to understand the properties of the distribution of growth probabilities on randomly generated sets, which are given by harmonic measures. The problem of estimating the harmonic measure of sets of points is addressed in this book, and some bounds on the measure derived. These, for example, give a lower bound on the fractal dimension of the diffusion-limited aggregates.

If the set of stopping points of walker is itself an infinite set, say generated as the sites visited by another walker, this becomes related to the question as to how often the paths of these two walkers intersect. In this case the basic result is that in more than 4 dimensions there is a finite probability that paths of two walkers starting at the same point will never intersect again, however long we wait. In less than 4 dimensions, the probability that no intersection of paths occurs within L steps decreases as a power of L Lawler discusses bound on this no-intersection probability. Another related subject discussed is the study of self-intersection of random walks. In a chapter devoted to self-avoiding walks, it is shown that the Edwards model is not in the same universality class as the self-avoiding walks in two dimensions. One chapter in the book is devoted to loop-erased walks. Much of what is known about loop-erased walks is due to Lawler, and this is the most readable account of this subject that I am aware of.

This monograph contains a clear account of these topics, and there is an emphasis on rigorous results. The book has an extensive bibliography. It is likely to interest a smaller set of people than the book by Madras and Slade. but it will be useful to graduate students and researchers who study random walks from a mathematician's viewpoint, and need to go beyond the standard textbooks, such as those by Feller and Spitzer.

1.4. Review by: Richard T Durrett.
SIAM Review 34 (4) (1992), 666.

Ever since Polya took a walk one Sunday afternoon at the turn of the century in Zurich, simple random walk has been an inspiration for mathematicians and a useful component of models in a number of applications. ...
...
I think anyone who reads the book will be struck by the difficulty of the simply stated problems and the ingenuity of their not yet complete solutions. Much of the material is Lawler's own research, so he knows his story thoroughly and he tells it well.

1.5. Review by: R E Bradley.
The Annals of Probability 24 (3) (1996), 1638-1642.

Random walks have fascinated and perplexed the mathematical community for about a century. Although there are a variety of complications and variations by means of which the basic model can be generalised, the behaviour in the simplest case is already complex and surprising.
...
Lawler gratefully acknowledges his debt to Spitzer's book for his exposition of the local central limit theorem, as well as for other standard results which are proved in the opening chapter. The intended audience of this book would appear to be upper level graduate students and working research mathematicians because so many standard results from measure theory and a year of graduate level probability are assumed without proof: for example, martingales, stopping times and Brownian motion. Beyond this, the text is almost entirely self-contained, with the exception of certain routine extensions which are included among the exercises (there are some 15 exercises included in the first two chapters).
...
It would only be a modest exaggeration to call this book a gem. It is relatively short and has a unity of purpose, both of which serve to give it a manageable scale. It is very technical, especially in Chapters 2-5, but the organisation is efficient without descending into slickness. Most of the results are Lawler's own, drawn from a series of at least eight articles published between 1980 and 1990; it is far more satisfying to review these results as a coherent and well-organised whole than in a piecemeal fashion. This book should be considered almost indispensable for those with an interest in self-avoiding walks and will appeal to anyone interested in applications of probability theory and harmonic measure.

1.6. Review by: Krzysztof Burdzy.
Mathematical Reviews MR1117680 (92f:60122).

Some of the hardest and most intriguing problems in probability come from analysing the self-avoiding random walk. Consider a uniform measure on all n-step non-self-intersecting simple (i.e., nearest neighbour) random walk paths on $\mathbb{Z}^{d}, d ≥ 2$. A path chosen according to this measure is called a self-avoiding random walk of length n. What is the average diameter of a self-avoiding random walk path of length n? This and related questions lead to "non-intersection" problems, i.e., problems involving estimates of the probability that several independent paths do not intersect. Three main chapters of the book present the best known rigorous estimates for the probability of non-intersection of several independent simple random walk paths. Special attention is paid to dimensions $d$ = 2, 3 and 4 because the results for $d ≤ 4$ differ qualitatively from those for higher dimensions.

A less formal chapter presents a review of self-avoiding random walks. Another chapter provides information on the loop-erased random walk, i.e., a model dealing with non-self-intersecting paths obtained by erasing loops of the simple random walk.

The first two chapters contain a review of results on random walks which may also be of interest to general audiences. One of their sections is devoted to diffusion limited aggregation (DLA) - a little understood model for fractal growth.

Although the technical part of the book is focused on non-intersection estimates, many aspects of the field are reviewed (e.g., Monte Carlo simulation techniques, non-rigorous estimates). The book has a good chance of becoming one of the standard references in the field for a long time as rapid progress in the area seems unlikely without the emergence of significantly different techniques.

2.1. From the Publisher.

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-dimensional.

Introduction 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.

2.2. From the Preface.

This book is an outgrowth of lectures in Mathematics 240, "Applied Stochastic Processes," which I have taught a number of times at Duke University. The majority of the students in the course are graduate students from departments other than mathematics, including computer science, economics, business, biological sciences, psychology, physics, statistics, and engineering. There have also been graduate students from the mathematics department as well as some advanced undergraduates. The mathematical background of the students varies greatly, and the particular areas of stochastic processes that are relevant for their research also vary greatly.

The prerequisites for using this book are a good calculus-based undergraduate course in probability and a course in linear algebra including eigenvalues and eigenvectors. I also assume that the reader is reasonably computer literate. The exercises assume that the reader can write simple programs and has access to some software for linear algebra computations. In all of my classes, students have had sufficient computer experience to accomplish this. Most of the students have also had some exposure to differential equations and I use such ideas freely, although I have a short section on linear differential equations in the preliminary chapter.

I have tried to discuss key mathematical ideas in this book, but I have not made an attempt to put in all the mathematical details. Measure theory is not a prerequisite but I have tried to present topics in a way such that readers who have some knowledge of measure theory can fill in details. Although this is a book intended primarily for people with applications in mind, there are few real applications discussed. True applications require a good understanding of the field being studied and it is not a goal of this book to discuss the many different fields in which stochastic processes are used. I have instead chosen to stick with the very basic examples and let the experts in other fields decide when certain mathematical assumptions are appropriate for their application.

Chapter 1 covers the standard material on finite Markov chains. I have not given proofs of the convergence to equilibrium but rather have emphasised the relationship between the convergence to equilibrium and the size of the eigenvalues of the stochastic matrix. Chapter 2 deals with infinite state space. The notions of transience, null recurrence, and positive recurrence are introduced, using as the main example, a random walk on the nonnegative integers with reflecting boundary. The chapter ends with a discussion of branching processes.

Continuous-time Markov chains are discussed in Chapter 3. The discussion centres on three main types: Poisson process, finite state space, and birth-and-death processes. For these processes I have used the forward differential equations to describe the evolution of the probabilities. This is easier and more natural than the backward equations. Unfortunately, the forward equations are not a legitimate means to analyse all continuous-time Markov chains and this fact is discussed briefly in the last section. One of the main examples of a birth-and-death process is a Markovian queue.

I have included Chapter 4 on optimal stopping of Markov chains as one example in the large area of decision theory. Optimal stopping has a nice combination of theoretical mathematics leading to an algorithm to solve a problem. The basic ideas are also similar to ideas presented in Chapter 5.

The idea of a martingale is fundamental in much of stochastic processes, and the goal of Chapter 5 is to give a solid introduction to these ideas. The modern definition of conditional expectation is first discussed and the idea of "measurable with respect to $\mathcal F_n$, the information available at time $n$" is used freely without worrying about giving it a rigorous meaning in terms of $\sigma$-algebras. The major theorems of the area, optional sampling and the martingale convergence theorem, are discussed as well as their proofs. Proofs are important here since part of the theory is to understand why the theorems do not always hold. I have included a discussion of uniform integrability.

The basic ideas of renewal theory are discussed in Chapter 6. For non-lattice random variables the renewal equation is used as the main tool of analysis while for lattice random variables a Markov chain approach is used. As an application, queues with general service times are analysed.

Chapter 7 discusses a couple of current topics in the realm of reversible Markov chains. First a more mathematical discussion about the rate of convergence to equilibrium is given, followed by a short introduction to the idea of Markov chain algorithms which are becoming very important in some areas of physics, computer science, and statistics. The final section on recurrence is a nice use of "variational" ideas to prove a result that is hard to prove directly.

Chapter 8 gives a very quick introduction to a large number of ideas in Brownian motion. It is impossible to make any attempt to put in all the mathematical details. I have discussed multidimensional as well as one-dimensional Brownian motion and have tried to show why Brownian motion and the heat equation are basically the same subject. I have also tried to discuss a little of the fractal nature of some of the sets produced by Brownian motion. In Chapter 9, a very short introduction to the idea of stochastic integration is given. This also is a very informal discussion but is intended to allow the students to at least have some ideas of what a stochastic integral is.

This book has a little more than can be covered in a one semester course. In my view the basic course consists of Chapters 1, 2, 3, 5, and 8. Which of the remaining chapters I cover depends on the particular students in the class that semester. The basic chapters should probably be done in the order listed, but the other chapters can be done at any time. Chapters 4 and 7 use the previous material on Markov chains; Chapter 6 uses Markov chains and martingales in the last section; and Chapter 9 uses the definition of Brownian motion as well as martingales.

2.3. Review by: Eric S Key.
SIAM Review 40 (1) (1998), 159-160.

Introduction to Stochastic Processes is a concise and thoroughly readable treatment of the subject aimed at students who are comfortable with elementary calculus, linear algebra, and linear differential equations and who have had a calculus-based undergraduate course in probability. Roughly half of the book is devoted to the study of various types of Markov chains. To get things going, Lawler begins with the simplest case, finite state chains, where he develops the ideas of invariant distributions, classification of states, and return times. He then turns to countably infinite state chains and discusses positive and null recurrence. This chapter culminates in a discussion of single type branching processes.
...
With the exception of the chapter on martingales, the style of presentation is to give definitions, examples, and statements of theorems. In general theorems are not proven, but the examples are sufficient to convince the reader of their validity. As the intended audience for this text is not graduate students in the mathematical sciences, this style seems ideal.
...
I think Lawler's book would suit an audience [in the biological sciences and social sciences] perfectly and would provide an excellent foundation for a course in stochastic processes for such a group of students.

2.4. Review by: A V Metcalfe.
Journal of the Royal Statistical Society. Series D (The Statistician) 45 (4) (1996), 533.

The third title in the Chapman and Hall 'Probability' series is this clearly written introduction to stochastic processes. Although the intended readership includes people who wish to apply the theory, the author has chosen to concentrate on the underlying mathematical concepts rather than practical applications. It is assumed that students of engineering, science and social sciences will be motivated by practical cases from their own discipline. The many examples in the book are used to explain the mathematical ideas.

A brief preliminary chapter summarises the key results for solving linear differential and difference equations, as a system of first-order equations in matrix terms. The first two main chapters cover Markov chains with finite and infinite state spaces. The second includes queues, random walks and population growth with discrete time steps. In Chapter Three these ideas are extended to continuous time and include Poisson processes, birth-and-death processes and Markovian queuing models. The basic ideas of optimal stopping of Markov chains are described in the short chapter which follows.

Chapter Five is an introduction to martingales. It includes careful discussions of the conditions under which the optimal sampling and martingale convergence theorems hold. The subject of queuing is taken up again in Chapter Six on renewal processes. The service, or arrival, times can now have a general distribution. A discussion of reversible Markov chains follows in Chapter Seven. The application to Monte Carlo simulation, which includes the Ising model as an example, is brief but informative. The last two chapters contain an exceptionally clear, and intuitive, account of Brownian motion and stochastic integration, including Ito's formula.

This book is a succinct introduction to stochastic processes which is ideal for private study, providing a sound mathematical basis for applications or more advanced work. As such, it fills a significant gap in the textbook market. It would also be an excellent text for a first course in stochastic processes. The prerequisite is some knowledge of probability, calculus and linear algebra.

2.5. Review by: P E.
Journal of the American Statistical Association 90 (432) (1995), 1493.

This is a fairly standard introduction to applied stochastic processes, requiring only a good calculus-based undergraduate course in probability and a course in linear algebra including eigenvalues and eigenvectors as preliminaries. Key topics are Markov chains, in both discrete and continuous time. A short excursion toward Brownian motion and stochastic integration opens the door for more advanced theory. Well-chosen examples and interesting exercises help make this text a good choice for a first course in stochastic processes for a broad class of students.

2.6. Review by: Anon.
Biometrics 53 (2) (1997), 783.

This is a beginner's text on stochastic processes aimed at American graduate students who have done "a good calculus-based course in probability theory and a course in linear algebra including eigenvalues and eigenvectors". The exercises assume that the reader is able to write simple computer programs and can use software for linear algebra computations. A course in differential equations would be advantageous, though there is a short section on them in the preliminary chapter. Similarly, a knowledge of measure theory is not an essential prerequisite, though it would be helpful.
...
The graduates that the author has primarily in mind are those from disciplines other than mathematics. Accordingly, in his discussion of key mathematical ideas he has not attempted "to put in all the mathematical details". Also, although the audience may have applications uppermost in mind, there is little discussion of real applications. The book is a friendly and clearly written introductory text with an abundance of material (perhaps a little too much for one semester). There are plenty of exercises, but no answers and no suggestions for further reading.

3.1. From the Publisher.

Emphasising 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 Itô'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.

3.2. From the Preface.

In the second edition we have significantly expanded the chapter on stochastic integration in order to give an introduction to modern mathematical finance. We have expanded the discussion of Itô's formula, introduced the Girsanov transformation and the Feynman-Kac formula, and derived the Black-Scholes formula for pricing options. We have tried to present this material in the same styles as other topics, that is, without complete mathematical details, but with enough ideas to explain to the reader why formulas are true.

We have added a section on maximal inequalities to the martingale section and included more material on Brownian motion. We have included a few more examples throughout the book and have increased the number of exercises at the end of the chapters. We have also made corrections and minor revisions in many places and included some recommendations for further reading.

3.3. Review by: Emmanuelle Clement.
Mathematical Reviews MR2255511 (2008d:60001).

The first edition appeared in 1995.

This textbook provides a clearly written introduction to stochastic processes which can be useful for students in many scientific fields. The prerequisites for reading this book are only a good calculus-based undergraduate course in probability and a course in linear algebra. Measure theory is not required, and therefore this book is well suited to a first course in stochastic processes. The key mathematical ideas are discussed through many basic examples and facts are derived emphasising the mathematical intuition rather than rigorous proofs or mathematical details. Each chapter contains many exercises.

Chapters one and two cover Markov chains with finite and countable state spaces. Chapter three presents continuous time Markov chains through three examples: Poisson process, finite state space and birth-and-death processes. Optimal stopping is described in chapter four. Chapter five is an introduction to martingales with the notion of conditional expectation. A section on maximal inequalities has been added to this edition. Renewal theory is introduced in chapter six and chapter seven gives some results on reversible Markov chains. Chapter eight is an introduction to Brownian motion, multidimensional as well as one-dimensional. Chapter nine is devoted to stochastic integration. This chapter has been expanded in the second edition in order to present the tools of modern mathematical finance, including Itô's formula, Girsanov transformation, and the Feynman-Kac formula as well as the Black-Scholes formula.

4.1. From the Publisher.

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.

4.2. From the Preface.

These notes summarise two of the three classes held for undergraduates at the 1996 Park City/ IAS Institute in Probability. There were twenty undergraduates participating, who were divided into an advanced group and a beginning group. Both groups participated in a class on computer simulations in probability. The beginner's class, taught by Emily Puckette, discussed Markov chains and random walks .... This book gives notes from the advanced class and the computer class. The first ten lectures are those given to the advanced class by Greg Lawler, and the last three summarise the material in the computer class led by Lester Coyle. The material was coordinated so that some of the major simulations done in the computer class related to topics discussed in the advanced class. For this reason we have decided to combine these notes into one book.

The title of the advanced lecture series (and of these notes) is taken from a recent book, Topics in Contemporary Probability and its Applications, edited by J Laurie Snell, which contains a number of survey articles that are accessible to advanced undergraduates and beginning graduate students. The lectures were based loosely on three of the papers in that book: "Random Walks: Simple and Self-Avoiding" by Greg Lawler, "How Many Times Should You Shuffle a Deck of Cards?" by Brad Mann, and "Uniform Spanning Trees" by Robin Pemantle. The idea was to present some topics which are accessible to advanced undergraduates yet are areas of current research in probability.

The first lecture discusses simple random walk in one dimension and is anything but contemporary. It leads to a derivation of Stirling's formula. The second lecture discusses random walk in several dimensions and introduces the notion of power laws. Standard results about probability of return to the origin and the intersection exponent are discussed. The latter is a simply stated exponent whose value is not known rigorously today, and it is a natural exponent to study by simulation. The third lecture discusses the self-avoiding walk, which is a very good example of a simply stated mathematical problem for which most of the interesting questions are still open problems. The fourth lecture considers the continuous limit of random walk, Brownian motion. This topic was included to help those students who were involved in simulations related to finance.

The next two lectures consider the problem of shuffling a deck of cards. Lecture 5 discusses the general idea of random permutations and introduces the notion of a random walk on a symmetric group. The case of random riffle shuffles and the time (number of shuffles) needed to get close to the uniform distribution was analysed in a paper of Bayer and Diaconis, and Mann's paper is an exposition of this result. We give a short discussion of this result, although we do not give all the details of the proof. This topic leads naturally to the discussion of Markov chains and rates of convergence to equilibrium. Lecture 7 is a standard introduction to Markov chains; it outlines a proof of convergence to equilibrium that emphasises the importance of the size of the second eigenvalue for understanding the rate of convergence. Lecture 8 discusses a recent important technique to sample from complicated distributions, Markov Chain Monte Carlo.

Lecture 9 discusses a very beautiful relationship between random walks and electrical networks. The basic ideas in this section are used in more sophisticated probability; this is basically the discrete version of Dirichlet forms. The work on electrical networks leads to the final lecture on uniform spanning trees. We discuss one result that relates three initially quite different objects: uniform spanning trees, random walks on graphs, and electrical networks.

The purpose of the computer class was to introduce students to the idea of Monte Carlo simulations and to give them a chance to do some nontrivial projects. The previous computer experience of the students varied widely, some having significant programming backgrounds and some having never computed. We first used Maple and then C as the languages for simulations. While these sections are labelled as "lectures" they actually represent a summary of many lectures, and the topics were not really presented in the order that they appear here. Lecture 11 discusses simulations for random walks and includes some basic material on curve fitting to estimate exponents. It ends with a discussion of the most serious project done in this area, the estimate of the intersection exponent. Lecture 12 discusses simulation topics other than random walk that were discussed in the class, including sampling from continuous distributions, random permutations, and finally a more difficult project - using Markov Chain Monte Carlo as discussed in Lecture 8 to estimate the number of matrices with certain conditions. The last lecture discusses a different area, simulations of stochastic differential equations for applications in finance.

We conclude the book with a number of problems that were presented to the students. The difficulty of these problems varies greatly: some are routine, but many were given more to stimulate thought than with the expectation that the students would completely solve them. They are numbered to indicate which lecture they refer to. Of particular note are the problems from Lectures 11 and 12. These are representative of the simpler projects that we gave to the students as they were learning how to do simulations, and are typical of simulation problems that we give to students when we teach undergraduate probability.

4.3. Review by: M T W.
Journal of the American Statistical Association 95 (450) (2000), 689.

This nice, short monograph contains material from lectures and computer labs held at the lAS/ Park City Mathematics Institute. The lectures present areas of modem probability theory that are current area s of research. The material is accessible to undergraduate students with a modest background in probability. The lecture topics include random walks, Brownian motion, card shuffling, spanning trees, and Markov chain Monte Carlo. These are computer simulations for random walks, Markov chains, stochastic differential equations as applied to finance, and other topics. This collection of lectures will be an excellent supplement to any intermediate probability course.

5.1. From the Publisher.

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.

5.2. From the Preface.

A number of two-dimensional lattice models in statistical physics have continuum limits that are conformally invariant. For example, the limit of simple random walk is Brownian motion. This book will discuss the nature of conformally invariant limits. Most of the processes discussed in this book are derived in one way or another from Brownian motion. The exciting new development in this area is the Schramm-Loewner evolution (SLE), which can be considered as a Brownian motion on the space of conformal maps.

These notes arise from graduate courses at Cornell University given in 2002-2003 on the mathematics behind conformally invariant processes. This may be considered equal doses of probability and conformal mapping. It is assumed that the reader knows the equivalent of first-year graduate courses in real analysis, complex analysis, and probability.

Here is an outline of the book. We start with a quick introduction to some discrete processes which have scaling limits that are conformally invariant. We only present enough here to whet the appetite of the reader, and we will not use this section later in the book. We will not prove any of the important results concerning convergence of discrete processes. ...

Chapter 1 gives the necessary facts about one-dimensional Brownian motion and stochastic calculus. We have given an essentially self-contained treatment; in order to do so, we only integrate with respect to continuous semi-martingales derived from Brownian motion and we only integrate adapted processes that are continuous (or piecewise continuous). The latter assumption is more restrictive than one generally wants for other applications of stochastic calculus, but it suffices for our needs and avoids having to discuss certain technical aspects of stochastic calculus. ... Sections 1.10 and 1.11 discuss some particular stochastic differential equations that arise in the analysis of SLE. The reader may wish to skip these sections until Chapter 6 where these equations appear; however, since they discuss properties of one-dimensional equations it logically makes sense to include them in the first chapter.

The next chapter introduces the basics of two-dimensional (i.e., one complex dimension) Brownian motion. It starts with the basic fact (dating back to Levy and implicit in earlier work on harmonic functions) that complex Brownian motion is conformally invariant. Here we collect a number of standard facts about harmonic functions and Green's function for complex Brownian motion. Because much of this material is standard, a number of facts are labelled as exercises.

Conformal mapping is the topic of Chapter 3. The purpose is to present the material about conformal mapping that is needed for SLE, especially material that would not appear in a first course in complex variables. ... However, our treatment here differs in some ways, most significantly in that it freely uses Brownian motion. We start with simple connectedness and a proof of the Riemann mapping theorem. Although this is really a first-year topic, it is so important to our discussion that it is included here. ... We then discuss two kinds of capacity, logarithmic capacity in the plane which is classical and a "half-plane" capacity that is not as well known but similar in spirit. Important uniform estimates about certain conformal transformations are collected here; these are the basis for Loewner differential equations. Extremal distance (extremal length) is an important conformally invariant quantity and is discussed in Section 3.7. The next section discusses the Beurling estimate, which is a corollary of a stronger result, the Beurling projection theorem. This is used to derive a number of estimates about conformal maps of simply connected domains near the boundary; what makes this work is the fact that the boundary of a simply connected domain is connected. The final section discusses annuli, which are important when considering radial or whole-plane processes.

Chapter 4 discusses the Loewner differential equation. We discuss three types, chordal, radial, and whole-plane, although the last two are essentially the same. It is the radial or whole-plane version that Loewner developed in trying to study the Bieberbach conjecture and has become a standard technique in conformal mapping theory. The chordal version is less well known; Schramm naturally came upon this equation when trying to find a continuous model for loop-erased walks and percolation. The final three sections deal with technical issues concerning the equation. When does the solution of the Loewner equation come from a path? What happens when solutions of the Loewner equation are mapped by a conformal transformation? What does it mean for a sequence of solutions of the Loewner equation to converge? The second of these questions is relevant for understanding the relationship between the chordal and radial Loewner equations.

In Chapter 5 we return to Brownian motion. Some of the most important conformally invariant measures on paths are derived from complex Brownian motion. After discussing a number of well-known measures (with perhaps a slightly different view than usual), we discuss some important measures that have arisen recently: excursion measure, Brownian boundary bubble measure, and the loop measure.
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5.3. Review by: Zhen-Qing Chen.
Mathematical Reviews MR2129588 (2006i:60003).

Theoretical physicists have predicted that the scaling limits of many two-dimensional lattice models in statistical physics are conformally invariant. Recently there has been remarkable and exciting progress, mainly made in joint works by the author of this book together with O Schramm and W Werner, and by S Smirnov, in obtaining mathematically rigorous proofs of the existence and identification of such conformal invariant limits for several lattice models, including the loop-erased random walk, the uniform spanning tree Peano curve and the exploration process for critical percolation on a triangular lattice. These conformally invariant limits are described by Schramm-Löwner evolutions (SLE), which are powered by a one-parameter family of scaled one-dimensional Brownian motions.

This book gives a nice and systematic introduction to the continuous time conformally invariant processes in the plane, assuming only knowledge of first year graduate real analysis, complex analysis and probability theory.
...
This book is very well written, and can also be used as a graduate textbook for a topic course on SLE. The book focuses on basic properties of conformally invariant processes and their application to Brownian motion intersection exponents.

6.1. From the Publisher.

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 modelling

6.2. From the Preface.

Random walk - the stochastic process formed by successive summation of independent, identically distributed random variables - is one of the most basic and well-studied topics in probability theory. For random walks on the integer lattice $\mathbb{Z}^{d}$, the main reference is the classic book by Spitzer. This text considers only a subset of such walks, namely those corresponding to increment distributions with zero mean and finite variance. In this case, one can summarise the main result very quickly: the central limit theorem implies that under appropriate rescaling the limiting distribution is normal, and the functional central limit theorem implies that the distribution of the corresponding path-valued process (after standard rescaling of time and space) approaches that of Brownian motion.

Researchers who work with perturbations of random walks, or with particle systems and other models that use random walks as a basic ingredient, often need more precise information on random walk behaviour than that provided by the central limit theorems. In particular, it is important to understand the size of the error resulting from the approximation of random walk by Brownian motion. For this reason, there is need for more detailed analysis. This book is an introduction to the random walk theory with an emphasis on the error estimates. Although "mean zero, finite variance" assumption is both necessary and sufficient for normal convergence, one typically needs to make stronger assumptions on the increments of the walk in order to get good bounds on the error terms.

This project embarked with an idea of writing a book on the simple, nearest neighbour random walk. Symmetric, finite range random walks gradually became the central model of the text. This class of walks, while being rich enough to require analysis by general techniques, can be studied without much additional difficulty. In addition, for some of the results, in particular, the local central limit theorem and the Green's function estimates, we have extended the discussion to include other mean zero, finite variance walks, while indicating the way in which moment conditions influence the form of the error.

6.3. Review by: Andrew R Wade.
Mathematical Reviews MR2677157 (2012a:60132).

Random walks are fundamental stochastic models with wide-ranging applications and a richly-developed theory. In the book under review, the random walks are time- and space-homogeneous, so that (in the discrete-time setting) the process is described by partial sums of independent, identically distributed random vectors. The state space is taken to be a lattice, typically the $d$-dimensional integer lattice $\mathbb{Z}^{d}$. The classes of random walks that are considered are restricted to those that are described by standard $d$-dimensional Brownian motion in the usual scaling limit: so at the very least the increment distribution has zero mean and finite variance, and typically stronger conditions are imposed, such as the increments being bounded and having symmetric distribution. The prototypical example is the symmetric simple random walk (SRW), although often the results in the book hold more generally.

The book is aimed at researchers and graduate students familiar with the basics of measure-theoretic probability theory.
...
The book admirably serves its purpose as an introduction to its branch of the theory of random walks, as well as an excellent reference for many of the key results of the theory. The first part of the book, Chapters 1-8, gives an attractive, modern presentation of mostly classical material; to modern readers disconcerted by the typesetting of Spitzer's classic book [op. cit.], this will be especially welcome. The presentation of the dyadic coupling in Chapter 7 is one of the few textbook treatments of this important technique. The second part of the book, Chapters 9-11, gives an interesting introduction to more recent topics, including some treated in greater detail by Lawler in [op. cit.]. The book under review contains a wealth of carefully organised and clearly presented information, and will be a valuable addition to the shelves of any random-walk enthusiast.

7.1. From the Publisher.

The heat equation can be derived by averaging over a very large number of particles. Traditionally, the resulting PDE is studied as a deterministic equation, an approach that has brought many significant results and a deep understanding of the equation and its solutions. By studying the heat equation by considering the individual random particles, however, one gains further intuition into the problem. While this is now standard for many researchers, this approach is generally not presented at the undergraduate level. In this book, Lawler introduces the heat equation and the closely related notion of harmonic functions from a probabilistic perspective.

The theme of the first two chapters of the book is the relationship between random walks and the heat equation. The first chapter discusses the discrete case, random walk and the heat equation on the integer lattice; and the second chapter discusses the continuous case, Brownian motion and the usual heat equation. Relationships are shown between the two. For example, solving the heat equation in the discrete setting becomes a problem of diagonalisation of symmetric matrices, which becomes a problem in Fourier series in the continuous case. Random walk and Brownian motion are introduced and developed from first principles. The latter two chapters discuss different topics: martingales and fractal dimension, with the chapters tied together by one example, a random Cantor set.

The idea of this book is to merge probabilistic and deterministic approaches to heat flow. It is also intended as a bridge from undergraduate analysis to graduate and research perspectives. The book is suitable for advanced undergraduates, particularly those considering graduate work in mathematics or related areas.

7.2. From the Preface.

The standard model for the diffusion of heat uses the idea that heat spreads randomly in all directions at some rate. The heat equation is a deterministic (non-random), partial differential equation derived from this intuition by averaging over the very large number of particles. This equation can and has been traditionally studied as a deterministic equation. While much can be said from this perspective, one also loses much of the intuition that can be obtained by considering the individual random particles.

The idea in these notes is to introduce the heat equation and the closely related notion of harmonic functions from a probabilistic perspective. Our starting point is the random walk which in continuous time and space becomes Brownian motion. We then derive equations to understand the random walk. This follows the modern approach where one tries to combine probabilistic and deterministic methods to analyse diffusion.

Besides the random/deterministic dichotomy, another difference in approach comes from choosing between discrete and continuous models. The first chapter of this book starts with discrete random walk and then uses it to define harmonic functions and the heat equations on the integer lattice. Here one sees that linear functions arise, and the deterministic questions yield problems in linear algebra. In particular, solutions of the heat equation can be found using diagonalisation of symmetric matrices.

The next chapter goes to continuous time and continuous space. We start with the Brownian motion which is the limit of random walk. This is a fascinating subject in itself and it takes a little work to show that it exists. We have separated the treatment into Sections 2.1 and 2.6. The idea is that the latter section does not need to be read in order to appreciate the rest of the chapter. The traditional heat equation and Laplace equation are found by considering the Brownian particles. Along the way, it is shown that the matrix diagonalisation of the previous chapter turns into a discussion of Fourier series.

The third chapter introduces a fundamental idea in probability, martingales, that is closely related to harmonic functions. The viewpoint here is probabilistic. The final chapter is an introduction to fractal dimension. The goal, which is a bit ambitious, is to determine the fractal dimension of the random Cantor set arising in Chapter 3.

This book is derived from lectures given in the Research Experiences for Undergraduates (REU) program at the University of Chicago. The REU is a summer program taken in part or in full by about eighty mathematics majors at the university. The students take a number of mini-courses and do a research paper under the supervision of graduate students. Many of the undergraduates also serve as teaching assistants for one of two other summer programs, one for bright junior high and high school students and another designed for elementary and high school teachers. The first two chapters in this book come from mini-courses in 2007 and 2008, and the last two chapters from a 2009 course.

The intended audience for these lectures was advanced undergraduate mathematics majors who may be considering graduate work in mathematics or a related area. The idea was to present probability and analysis in a more advanced way than found in undergraduate courses. I assume the students have had the equivalent of an advanced calculus (rigorous one variable calculus) course and some exposure to linear algebra. I do not assume that the students have had a course in probability, but I present the basics quickly. I do not assume measure theory, but I introduce many of the important ideas along the way, such as: Borel-Cantelli lemma, monotone and dominated convergence theorems, Borel measure, conditional expectation, etc. I also try to firm up the students' grasp of the advanced calculus throughout the book. For example, analysis of simple random walk leads to Stirling's formula whose proof uses Taylor's theorem with remainder.

It is hoped that this book will be interesting to undergraduates, especially those considering graduate studies, as well as to graduate students and faculty whose specialty is not probability or analysis. This book could be used for advanced seminars or for independent reading. There are a number of exercises at the end of each section. They vary in difficulty and some of them are at the challenging level that corresponds to summer projects for undergraduates at the REU.

7.3. Review by: Peter Mörters.
Mathematical Reviews MR2732325 (2012c:60002).

This beautiful little book is an introduction to some of the key ideas of probability written at an advanced undergraduate level. It is suitable for getting undergraduates interested in studying probability at the graduate level, and may also serve as an invitation for students (or maybe even researchers) working in a neighbouring area to have a taste of the probabilistic way of thinking. The main themes treated are random walk, harmonic functions, the heat equation, Brownian motion and martingales, and the emphasis is on the relations among these key concepts.

The book comprises four chapters: Chapter 1 introduces simple random walk on the integer lattice and its basic properties using only elementary calculus and linear algebra. Through the gambler's ruin problem we are invited to look at discrete boundary value problems and harmonic functions on subsets of the lattice, and eventually the discrete heat equation. With this material at hand we can cross the divide into the continuous world with ease and learn in Chapter 2 about Brownian motion, the Dirichlet problem and the continuous heat equation. Chapter 3 offers a self-contained course on martingale theory, which is well motivated by the studies of the previous chapters. A short final chapter introduces the notion of Hausdorff dimension. The highlight here is the calculation of the dimension of the percolation limit set, which combines martingale methods and basic Hausdorff dimension techniques.

This book is very well written, self-contained up to material from elementary calculus and basic linear algebra, and has plenty of interesting exercises. It is well suited for an advanced undergraduate course, a student seminar or as material for an undergraduate project.

8.1. From the Publisher.

The title "Random Explorations" has two meanings. First, a few topics of advanced probability are deeply explored. Second, there is a recurring theme of analysing a random object by exploring a random path.

This book is an outgrowth of lectures by the author in the University of Chicago Research Experiences for Undergraduate (REU) program in 2020. The idea of the course was to expose advanced undergraduates to ideas in probability research.

The book begins with Markov chains with an emphasis on transient or killed chains that have finite Green's function. This function, and its inverse called the Laplacian, is discussed next to relate two objects that arise in statistical physics, the loop-erased random walk (LERW) and the uniform spanning tree (UST). A modern approach is used including loop measures and soups. Understanding these approaches as the system size goes to infinity requires a deep understanding of the simple random walk so that is studied next, followed by a look at the infinite LERW and UST. Another model, the Gaussian free field (GFF), is introduced and related to loop measure. The emphasis in the book is on discrete models, but the final chapter gives an introduction to the continuous objects: Brownian motion, Brownian loop measures and soups, Schramm-Loewner evolution (SLE), and the continuous Gaussian free field. A number of exercises scattered throughout the text will help a serious reader gain better understanding of the material.

8.2. From the Preface.

This book is an outgrowth of lectures that I gave in the summer of 2020 as part of the Research Experiences for Undergraduates (REU) at the University of Chicago. The REU lectures are not intended to be standard courses but rather tastes of graduate and research level mathematics for advanced undergraduates. The title of the book can be interpreted in two ways. First, this is not a comprehensive survey of an area but rather a "random" sampling of some objects that arise in models in probability and statistical mechanics. The second meaning refers to a prevailing theme in many of these models. Random fields can be studied by exploration, that is, by traveling (perhaps randomly) through the field and observing what one has seen so far and using that to predict the parts that have not been observed.

In order to keep the material accessible to students who have not had graduate material, I have concentrated on discrete models where "measure theoretic" probability is not needed. The formal prerequisites for these notes are advanced calculus, linear algebra, and a calculus-based course in probability. It is also expected that students have sufficient mathematical maturity to understand rigorous arguments. While those are the only formal prerequisites, the intent of these lectures was to give a taste of research level mathematics and I allow myself to venture occasionally a bit beyond these prerequisites.

The first chapter introduces Markov chains and ideas that permeate the book. The focus is on transient chains or recurrent chains with "killing" for which there is a finite Green's function representing the expected number of visits to sites. The Green's function can be seen to be the inverse of an important operator, the Laplacian. Harmonic functions (functions whose Laplacian equal zero) and the determinant of the Laplacian figure prominently in the later chapters. We concentrate mainly on discrete time chains but we also discuss how to get continuous time chains by putting on exponential waiting times. A probabilistic approach dominates our treatment but much can be done purely from a linear algebra perspective. The latter approach allows measures on paths that take negative and complex values. Such path measures come up naturally in a number of models in mathematical physics although they are not emphasized much here.

Chapter 2 introduces an object that has been a regular part of my research. I introduced the loop-erased random walk (LERW) in my doctoral dissertation with the hope of getting a better understanding of a very challenging problem, the self-avoiding random walk (SAW). While the differences between the LERW and SAW have prevented the former from being a tool to solve the latter problem, it has proved to be a very interesting model in itself. One very important application is the relation between LERW and another model, the uniform spanning tree (UST). This relationship is most easily seen in an algorithm due to David Wilson to generate such trees.

Analysis of the loop-erasing procedure leads to consideration both of the loops erased and the LERW itself. Chapter 3 gives an introduction to loop measures and soups that arise from this. We view a collection of loops as a random field that is growing with time as loops are added. The distribution of the loops at time 1 corresponds to what is erased from loop-erased random walks. The loop soup at time $\large\frac{1}{2}\normalsize$ is related to the Gaussian free field (GFF). This chapter introduces the discrete time loop soup which is an interesting mathematical model in itself. This discrete model has characteristics of a number of fields in statistical mechanics. In particular, the distribution of the field does not depend on how one orders the elements, but to investigate the field one can order the sites and then investigate the field one site at a time. For this model, when one visits a site, one sees all the loops that visits that site. This "growing loop" model which depends on the order of the vertices turns out to be equivalent to an "unrooted loop soup" that does not depend on the order.

While we have used the generality of Markov chains for our setup, one of the most important chains is the simple random walk in the integer lattice. In order to appreciate paths and fields arising from random walk, it is necessary to understand the walk. Chapter 4 discusses the simple random walk on the lattice giving some more classical results that go beyond what one would normally see at an undergraduate level.

We return to the spanning tree in Chapter 5 and consider the infinite spanning tree in the integer lattice as a limit of spanning trees on finite subsets. Whether or not this gives an infinite tree or a forest (a collection of disconnected trees) depends on the dimension. We also give an example of duality on the integer lattice.

Another classical field is the topic of Chapter 6. The multivariate normal distribution is a well known construction and is the model upon which much of classical mathematical statistics, such as linear regression, is based. The (GFF) is an example of such a distribution where some geometry comes into the picture. Here we discuss the GFF coming from a Markov chain. The idea of exploration comes in again as one "samples" or "explores" the field at some sites and uses that to determine distributions at other sites. The global object is independent of the ordering of the vertices but the sampling rule is not. ...

In Chapter 7 we introduce some of the continuous models that arise as scaling limits. A proper treatment of this material would require more mathematical background than I am assuming so this should be viewed as an enticement to learn more. The scaling limits we discuss are: Brownian motion, Brownian loop soup, Schramm- Loewner evolution, and the continuous GFF.
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8.3. Review by Achim Klenke.
Math Semesterber 70 (2023), 211-213.

Greg Lawler has made a name for himself with his work on loop-erased random walk, conformal invariance and Schramm-Loewner evolution (SLE) and is one of the most renowned researchers in this field. His current book promises to provide an elementary introduction to the topic of loop-erased random walks and to at least touch on conformal invariance and SLE. This has been very well done, although here too "elementary" does not mean "simple", but simply refers to the fact that the reader needs no prior knowledge of measure theory.
...

Overall, I think the book is very successful and I am pleased that I was able to take the time to read the work in detail as part of this review.