This book presents fundamental relations between random walks on graphs and field theories of mathematical physics. Such relations have been explored for several decades and remain a rapidly developing research area in probability theory.

The main objects of study include Markov loops, spanning forests, random holonomies, and covers, and the purpose of the book is to investigate their relations to Bose fields, Fermi fields, and gauge fields. The book starts with a review of some basic notions of Markovian potential theory in the simple context of a finite or countable graph, followed by several chapters dedicated to the study of loop ensembles and related statistical physical models. Then, spanning trees and Fermi fields are introduced and related to loop ensembles. Next, the focus turns to topological properties of loops and graphs, with the introduction of connections on a graph, loop holonomies, and YangMills measure. Among the main results presented is an intertwining relation between merge-and-split generators on loop ensembles and Casimir operators on connections, and the key reflection positivity property for the fields under consideration.

Aimed at researchers and graduate students in probability and mathematical physics, this concise monograph is essentially self-contained. Familiarity with basic notions of probability, Poisson point processes, and discrete Markov chains are assumed of the reader.

Explores an area of probability inspired by constructive quantum field theory Presents a variety of objects relevant to field theory in the simple framework of graphs Concisely written and essentially self-contained

Autorentext
Yves Le Jan is a French mathematician working in Probability theory and Stochastic processes. In 2006 he was invited speaker at the International Congress of Mathematicians in Madrid. In 2008 he became Senior Member of the Institut Universitaire de France. In 2011 he was Doob Lecturer at the 8th World Congress in Probability and Statistics in Istanbul. In 1995 he was awarded the Poncelet Prize and in 2011 the Sophie Germain Prize of the French Academy of Sciences. He is Professor emeritus at the Orsay Institute of Mathematics of Université Paris-Saclay and since 2021 visiting Professor at NYUAD.

Zusammenfassung

I enjoyed reading this very well-written and interesting book. It introduces objects such as Markov loops, spanning forests, random homologies and covers, which are related to random walks on graphs. Relations between these objects and random fields used in mathematical physics are explored. (John Masson Noble, Mathematical Reviews, May, 2025)

Inhalt
1 Markov Chains and Potential Theory on Graphs.- 2 Loop Measures.- 3 Decompositions, Traces and Excursions.- 4 Occupation Fields.- 5 Primitive Loops, Loop Clusters, and Loop Percolation.- 6 The Gaussian Free Field.- 7 Networks, Ising Model, Flows, and Configurations.- 8 Loop Erasure, Spanning Trees and Combinatorial Maps.- 9 Fock Spaces, Fermi Fields, and Applications.- 10 Groups and Covers.- 11 Holonomies and Gauge Fields.- 12 Reflection Positivity and Physical Space.