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

Le Jan, Y.

Springer（2011/07発売）

• 製本 Paperback:紙装版/ペーパーバック版／ページ数 124 p.／サイズ 11 illus.
• 商品コード 9783642212154

Full 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.

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.