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Random Explorations
By Gregory F. Lawler. STML/98, 2022, 199 pp.
The main goal of this book is to explain to the reader several important topics in probability theory and related areas of statistical physics.
The author starts with a well-known class of Markov chains called random walks on graphs. In the first chapter he recalls the main notions of the theory of Markov chains with or without a boundary, and introduces the Laplacian, the Poisson kernel, and the Green function of a Markov chain.
In Chapter 2 the main character, called Loop Erasing Random Walk (LERW) enters the stage. True to its name, LERW is obtained from an ordinary random walk on a graph by erasing loops (which occur when a path returns to a graph vertex it already visited). The result of this loop erasure process is a path without self-intersections. The probability distribution on the set of non-self-intersecting paths obtained by pushing forward the original probability distribution on all paths is introduced and analyzed.
The loop erasing procedure applied to a random walk not only creates a non-self-intersecting path, but also generates a set of cut-off loops, called the loop soup. This set of loops comes with a probability distribution, and the main result in Chapter 3 shows how to attach loops back to non-self-intersecting paths in order to obtain the original probability distribution (random walk) on the set of all paths.
The most common random walks are random walks on subgraphs of the integer lattice in the Euclidean space, and Chapter 4 is devoted to such random walks, the corresponding Green function, harmonic functions, and the notion of capacity. In Chapter 5, LERWs on lattice graphs -dimensional are analyzed and their relationship to the so-called spanning trees (i.e., maximal subgraphs of that do not contain cycles) is studied. The presentation is given separately for where a typical path does not return to the starting point, and for , where it does. ,
In Chapter 6 the author introduces Gaussian Free Fields (GFF), first as general multinomial distributions, and then as the distribution related to the random walk on a graph. The Gibbs measure construction of Gaussian random fields is explained and the one-dimensional case is considered in detail. The final Chapter 7 is devoted to a broad overview of the notion of scaling limit in probability theory and statistical physics, which allows us to pass from discrete random models to continuous ones. This notion is illustrated by several examples such as passing from discrete random walks to Brownian motion, from discrete loop soup to continuous loop soup, from LERW to the Schramm–Loewner equation, and, finally, from discrete Gaussian fields to continuous Gaussian free fields. The general notion of conformal invariance in dimension 2 is also discussed in this chapter.
A very attractive feature of the book is that the reader does not need to know much to be able to read it: advanced calculus, linear algebra, and beginning probability (without measure theory) would suffice. A few additional notions and results, such as second moment methods in probability, the definition of the Gamma process, and the definition of the Levy process are explained in the Appendix. On the other hand, the reader should be motivated enough to work through rigorous mathematical definitions and proofs. Numerous examples will help the reader in her journey through the beautiful land of quantitative randomness ruled by the laws of probability theory.