LEADER 04047nam 22006375 450 001 9910869179403321 005 20260112105650.0 010 $a9783031579233$b(electronic bk.) 010 $z9783031579226 024 7 $a10.1007/978-3-031-57923-3 035 $a(MiAaPQ)EBC31512107 035 $a(Au-PeEL)EBL31512107 035 $a(CKB)32650735500041 035 $a(DE-He213)978-3-031-57923-3 035 $a(EXLCZ)9932650735500041 100 $a20240701d2024 u| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aRandom Walks and Physical Fields /$fby Yves Le Jan 205 $a1st ed. 2024. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2024. 215 $a1 online resource (188 pages) 225 1 $aProbability Theory and Stochastic Modelling,$x2199-3149 ;$v106 311 08$aPrint version: Le Jan, Yves Random Walks and Physical Fields Cham : Springer,c2024 9783031579226 320 $aIncludes bibliographical references and index. 327 $a1 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. 330 $aThis book presents fundamental relations between random walks on graphs and field theories of mathematical physics. Such relations have been explored for several decades and remain a rapidly developing research area in probability theory. The main objects of study include Markov loops, spanning forests, random holonomies, and covers, and the purpose of the book is to investigate their relations to Bose fields, Fermi fields, and gauge fields. The book starts with a review of some basic notions of Markovian potential theory in the simple context of a finite or countable graph, followed by several chapters dedicated to the study of loop ensembles and related statistical physical models. Then, spanning trees and Fermi fields are introduced and related to loop ensembles. Next, the focus turns to topological properties of loops and graphs, with the introduction of connections on a graph, loop holonomies, and Yang?Mills measure. Among the main results presented is an intertwining relation between merge-and-split generators on loop ensembles and Casimir operators on connections, and the key reflection positivity property for the fields under consideration. Aimed at researchers and graduate students in probability and mathematical physics, this concise monograph is essentially self-contained. Familiarity with basic notions of probability, Poisson point processes, and discrete Markov chains are assumed of the reader. 410 0$aProbability Theory and Stochastic Modelling,$x2199-3149 ;$v106 606 $aProbabilities 606 $aMathematical physics 606 $aParticles (Nuclear physics) 606 $aQuantum field theory 606 $aProbability Theory 606 $aMathematical Physics 606 $aElementary Particles, Quantum Field Theory 606 $aRutes aleatòries (Matemàtica)$2thub 608 $aLlibres electrònics$2thub 615 0$aProbabilities. 615 0$aMathematical physics. 615 0$aParticles (Nuclear physics) 615 0$aQuantum field theory. 615 14$aProbability Theory. 615 24$aMathematical Physics. 615 24$aElementary Particles, Quantum Field Theory. 615 7$aRutes aleatòries (Matemàtica) 676 $a519.282 700 $aLe Jan$b Yves$062931 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 912 $a9910869179403321 996 $aRandom Walks and Physical Fields$94170555 997 $aUNINA