Zurich Lectures in Advanced Mathematics 2012; 122 pp; softcover Volume: 16 ISBN10: 3037191090 ISBN13: 9783037191095 List Price: US$36 Member Price: US$28.80 Order Code: EMSZLEC/16
 This book grew out of a graduate course at ETH Zürich during the spring 2011 term. It explores various links between such notions as occupation times of Markov chains, Gaussian free fields, Poisson point processes of Markovian loops, and random interlacements, which have been the object of intensive research over the last few years. These notions are developed in the convenient setup of finite weighted graphs endowed with killing measures. This book first discusses elements of continuoustime Markov chains, Dirichlet forms, potential theory, together with some consequences for Gaussian free fields. Next, isomorphism theorems and generalized RayKnight theorems, which relate occupation times of Markov chains to Gaussian free fields, are presented. Markovian loops are constructed and some of their key properties derived. The field of occupation times of Poisson point processes of Markovian loops is investigated. Of special interest are its connection to the Gaussian free field, and a formula of Symanzik. Finally, links between random interlacements and Markovian loops are discussed, and some further connections with Gaussian free fields are mentioned. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Readership Graduate students and research mathematicians interested in occupation times, Gaussian free fields, Markovian loops, and random interlacements. Table of Contents  Introduction
 Generalities
 Isomorphism theorems
 The Markovian loop
 Poisson gas of Markovian loops
 References
 Index
