03779nam a22005295i 4500001001800000003000900018005001700027007001500044008004100059020003700100024003500137050001900172072001600191072001700207072002300224072001500247072001600262100008200278245007900360250001800439264006900457300004400526336002600570337002600596338003600622347002400658490006700682505048900749520143701238650002602675650002602701650004302727650003302770650003102803650002602834650005502860710003402915773002602949776003602975776003603011776003603047830007403083856004603157912001403203912001503217999001703232978-3-031-57923-3DE-He21320251030112344.0cr nn 008mamaa240701s2024    sz |    s    |||| 0|eng d  a97830315792339978-3-031-57923-37 a10.1007/978-3-031-57923-32doi 4aQA273.A1-274.9 7aPBT2bicssc 7aPBWL2bicssc 7aMAT0290002bisacsh 7aPBT2thema 7aPBWL2thema1 aLe Jan, Yves.eauthor.4aut4http://id.loc.gov/vocabulary/relators/aut97972410aRandom Walks and Physical Fieldsh[electronic resource] /cby Yves Le Jan.  a1st ed. 2024. 1aCham :bSpringer Nature Switzerland :bImprint: Springer,c2024.  aXIII, 184 p. 7 illus.bonline resource.  atextbtxt2rdacontent  acomputerbc2rdamedia  aonline resourcebcr2rdacarrier  atext filebPDF2rda1 aProbability Theory and Stochastic Modelling,x2199-3149 ;v1060 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.  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. 0aProbabilities.965724 0aMathematical physics. 0aElementary particles (Physics).968885 0aQuantum field theory.96888614aProbability Theory.96742824aMathematical Physics.24aElementary Particles, Quantum Field Theory.9688872 aSpringerLink (Online service)0 tSpringer Nature eBook08iPrinted edition:z978303157922608iPrinted edition:z978303157924008iPrinted edition:z9783031579257 0aProbability Theory and Stochastic Modelling,x2199-3149 ;v10697972540uhttps://doi.org/10.1007/978-3-031-57923-3  aZDB-2-SMA  aZDB-2-SXMS  c61200d61200