Le Jan, Yves.
Markov Paths, Loops and Fields École d'Été de Probabilités de Saint-Flour XXXVIII – 2008 / [electronic resource] : by Yves Le Jan. - Berlin, Heidelberg : Springer Berlin Heidelberg, 2011. - VIII, 124p. 9 illus. online resource. - Lecture Notes in Mathematics, 2026 0075-8434 ; . - Lecture Notes in Mathematics, 2026 .
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.
9783642212161
10.1007/978-3-642-21216-1 doi
Mathematics.
Potential theory (Mathematics).
Distribution (Probability theory).
Mathematics.
Probability Theory and Stochastic Processes.
Potential Theory.
QA273.A1-274.9 QA274-274.9
519.2
Markov Paths, Loops and Fields École d'Été de Probabilités de Saint-Flour XXXVIII – 2008 / [electronic resource] : by Yves Le Jan. - Berlin, Heidelberg : Springer Berlin Heidelberg, 2011. - VIII, 124p. 9 illus. online resource. - Lecture Notes in Mathematics, 2026 0075-8434 ; . - Lecture Notes in Mathematics, 2026 .
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.
9783642212161
10.1007/978-3-642-21216-1 doi
Mathematics.
Potential theory (Mathematics).
Distribution (Probability theory).
Mathematics.
Probability Theory and Stochastic Processes.
Potential Theory.
QA273.A1-274.9 QA274-274.9
519.2