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020 _a9783031579233
_9978-3-031-57923-3
024 7 _a10.1007/978-3-031-57923-3
_2doi
050 4 _aQA273.A1-274.9
072 7 _aPBT
_2bicssc
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072 7 _aMAT029000
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100 1 _aLe Jan, Yves.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_979724
245 1 0 _aRandom Walks and Physical Fields
_h[electronic resource] /
_cby Yves Le Jan.
250 _a1st ed. 2024.
264 1 _aCham :
_bSpringer Nature Switzerland :
_bImprint: Springer,
_c2024.
300 _aXIII, 184 p. 7 illus.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aProbability Theory and Stochastic Modelling,
_x2199-3149 ;
_v106
505 0 _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.
520 _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.
650 0 _aProbabilities.
_965724
650 0 _aMathematical physics.
650 0 _aElementary particles (Physics).
_968885
650 0 _aQuantum field theory.
_968886
650 1 4 _aProbability Theory.
_967428
650 2 4 _aMathematical Physics.
650 2 4 _aElementary Particles, Quantum Field Theory.
_968887
710 2 _aSpringerLink (Online service)
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9783031579226
776 0 8 _iPrinted edition:
_z9783031579240
776 0 8 _iPrinted edition:
_z9783031579257
830 0 _aProbability Theory and Stochastic Modelling,
_x2199-3149 ;
_v106
_979725
856 4 0 _uhttps://doi.org/10.1007/978-3-031-57923-3
912 _aZDB-2-SMA
912 _aZDB-2-SXMS
999 _c61200
_d61200