Name: Random Walks and Physical Fields
ISBN: 978-3-031-57922-6

Overview

Authors:

Yves Le Jan ⁰

Yves Le Jan
1. Laboratoire de mathématiques d'Orsay, University of Paris-Saclay, Orsay, France
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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

Part of the book series: Probability Theory and Stochastic Modelling (PTSM, volume 106)

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Keywords

Markov loops spanning trees
Graphs
Connections
Free field
Fock Spaces
Markov loop measures
Local times and free field
Ising model
Spanning trees
Fermi fields
Random holonomies
Casimir operator
Reflection positivity
Bose field

About this book

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 Yang–Mills 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.

Authors and Affiliations

Laboratoire de mathématiques d'Orsay, University of Paris-Saclay, Orsay, France

Yves Le Jan

About the author

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.

Bibliographic Information

Book Title: Random Walks and Physical Fields
Authors: Yves Le Jan
Series Title: Probability Theory and Stochastic Modelling
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2024
Hardcover ISBN: 978-3-031-57922-6Due: 04 August 2024
Softcover ISBN: 978-3-031-57925-7Due: 04 August 2024
eBook ISBN: 978-3-031-57923-3Due: 04 August 2024
Series ISSN: 2199-3130
Series E-ISSN: 2199-3149
Edition Number: 1
Number of Pages: X, 206
Number of Illustrations: 7 b/w illustrations

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Random Walks and Physical Fields