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Markov paths, loops and fields. École d’Été de Probabilités de Saint-Flour XXXVIII – 2008. (English) Zbl 1231.60002
Lecture Notes in Mathematics 2026. Berlin: Springer (ISBN 978-3-642-21215-4/pbk; 978-3-642-21216-1/ebook). viii, 124 p. (2012).
The purpose of this monograph is to investigate some relations between Markovian paths and loop measures, the Poissonian ensembles of loops they determine, their occupation fields, uniform spanning trees, determinants and Gaussian Markov fields. It contains the results of the author from [Ann. Probab. 38, No. 6, 1280–1319 (2010; Zbl 1197.60075)], where emphasis was on the study of occupation fields defined by Poissonian ensembles of Markov loops. The chapter headings are: 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.

MSC:
60-02 Research exposition (monographs, survey articles) pertaining to probability theory
60J27 Continuous-time Markov processes on discrete state spaces
60J45 Probabilistic potential theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
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