Markov Paths, Loops and Fields: École d'Été de Probabilités de Saint-Flour XXXVIII – 2008 (Lecture Notes in Mathematics Book 2026)

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. Front Matter....Pages i-viii Symmetric Markov Processes on Finite Spaces....Pages 1-12 Loop Measures....Pages 13-28 Geodesic Loops....Pages 29-34 Poisson Process of Loops....Pages 35-45 The Gaussian Free Field....Pages 47-56 Energy Variation and Representations....Pages 57-64 Decompositions....Pages 65-73 Loop Erasure and Spanning Trees....Pages 75-89 Reflection Positivity....Pages 91-97 The Case of General Symmetric Markov Processes....Pages 99-113 Back Matter....Pages 115-124