Markov Paths, Loops and Fields École d'Été de Probabilités de Saint-Flour XXXVIII – 2008 /

• Main Author: Le Jan, Yves (Author)
• Corporate Author: SpringerLink (Online service)
• Format: Electronic eBook
• Language: English
• Published: Berlin, Heidelberg : Springer Berlin Heidelberg, 2011.
• Series: Lecture Notes in Mathematics, 2026
• Subjects: Mathematics.; Potential theory (Mathematics).; Probabilities.; Probability Theory and Stochastic Processes.; Potential Theory.

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 studi...

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.