Markov paths, loops and fields
Author(s)
Bibliographic Information
Markov paths, loops and fields
（Lecture notes in mathematics, 2026 . Ecole d'été de probabilités de Saint-Flour ; 38-2008）
Springer, c2011
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Library, Research Institute for Mathematical Sciences, Kyoto University数研
L/N||LNM||2026200021321275
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Note
Includes bibliographical references (p. 115-117) and index
Description and Table of Contents
Description
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 fre 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.
Table of 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.
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