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Markov Paths, Loops and Fields [electronic resource] : École d'Été de Probabilités de Saint-Flour XXXVIII – 2008 / by Yves Le Jan.

By: Le Jan, Yves [author.].
Contributor(s): SpringerLink (Online service).
Material type: materialTypeLabelBookSeries: Lecture Notes in Mathematics: 2026Publisher: Berlin, Heidelberg : Springer Berlin Heidelberg, 2011Description: VIII, 124p. 9 illus. online resource.Content type: text Media type: computer Carrier type: online resourceISBN: 9783642212161.Subject(s): Mathematics | Potential theory (Mathematics) | Distribution (Probability theory) | Mathematics | Probability Theory and Stochastic Processes | Potential TheoryDDC classification: 519.2 Online resources: Click here to access online
Contents:
1 Symmetric Markov processes on finite spaces -- 2 Loop measures -- 3 Geodesic loops -- 4 Poisson process of loops -- 5 The Gaussian free field -- 6 Energy variation and representations -- 7 Decompositions -- 8 Loop erasure and spanning trees -- 9 Reflection positivity -- 10 The case of general symmetric Markov processes.
In: Springer eBooksSummary: The purpose of these notes is to explore some simple relations between Markovian path and loop measures, the Poissonian ensembles of loops they determine, their occupation fields, uniform spanning trees, determinants, and Gaussian Markov fields such as the free field. These relations are first studied in complete generality for the finite discrete setting, then partly generalized to specific examples in infinite and continuous spaces.
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1 Symmetric Markov processes on finite spaces -- 2 Loop measures -- 3 Geodesic loops -- 4 Poisson process of loops -- 5 The Gaussian free field -- 6 Energy variation and representations -- 7 Decompositions -- 8 Loop erasure and spanning trees -- 9 Reflection positivity -- 10 The case of general symmetric Markov processes.

The purpose of these notes is to explore some simple relations between Markovian path and loop measures, the Poissonian ensembles of loops they determine, their occupation fields, uniform spanning trees, determinants, and Gaussian Markov fields such as the free field. These relations are first studied in complete generality for the finite discrete setting, then partly generalized to specific examples in infinite and continuous spaces.

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