03994nam a22005775i 4500 978-3-031-57923-3 DE-He213 20251030112344.0 cr nn 008mamaa 240701s2024 sz | s |||| 0|eng d 9783031579233 978-3-031-57923-3 10.1007/978-3-031-57923-3 doi QA273.A1-274.9 PBT bicssc PBWL bicssc MAT029000 bisacsh PBT thema PBWL thema Le Jan, Yves. author. aut http://id.loc.gov/vocabulary/relators/aut 79724 Random Walks and Physical Fields [electronic resource] / by Yves Le Jan. 1st ed. 2024. Cham : Springer Nature Switzerland : Imprint: Springer, 2024. XIII, 184 p. 7 illus. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Probability Theory and Stochastic Modelling, 2199-3149 ; 106 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. 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. Probabilities. 65724 Mathematical physics. Elementary particles (Physics). 68885 Quantum field theory. 68886 Probability Theory. 67428 Mathematical Physics. Elementary Particles, Quantum Field Theory. 68887 SpringerLink (Online service) Springer Nature eBook Printed edition: 9783031579226 Printed edition: 9783031579240 Printed edition: 9783031579257 Probability Theory and Stochastic Modelling, 2199-3149 ; 106 79725 https://doi.org/10.1007/978-3-031-57923-3 ZDB-2-SMA ZDB-2-SXMS 61200 61200