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Markov Paths, Loops and Fields

École d'Été de Probabilités de Saint-Flour XXXVIII – 2008

  • Book
  • © 2011

Overview

Authors:
  1. Yves Le Jan

    1. CNRS, Département de Mathématiques, Université Paris-Sud, Orsay CX, France

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  • St Flour Lecture notes+ additional material.
  • A large part of the content is new.
  • There is no really similar book.
  • Includes supplementary material: sn.pub/extras

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2026)

Part of the book sub series: École d'Été de Probabilités de Saint-Flour (LNMECOLE)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-viii
    Download chapter PDF

  2. Symmetric Markov Processes on Finite Spaces

    • Yves Le Jan
    Pages 1-12

  3. Loop Measures

    • Yves Le Jan
    Pages 13-28

  4. Geodesic Loops

    • Yves Le Jan
    Pages 29-34

  5. Poisson Process of Loops

    • Yves Le Jan
    Pages 35-45

  6. The Gaussian Free Field

    • Yves Le Jan
    Pages 47-56

  7. Energy Variation and Representations

    • Yves Le Jan
    Pages 57-64

  8. Decompositions

    • Yves Le Jan
    Pages 65-73

  9. Loop Erasure and Spanning Trees

    • Yves Le Jan
    Pages 75-89

  10. Reflection Positivity

    • Yves Le Jan
    Pages 91-97

  11. The Case of General Symmetric Markov Processes

    • Yves Le Jan
    Pages 99-113

  12. Back Matter

    Pages 115-124
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Keywords

About this book

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.

Authors and Affiliations

  • CNRS, Département de Mathématiques, Université Paris-Sud, Orsay CX, France

    Yves Le Jan

Bibliographic Information

  • Book Title: Markov Paths, Loops and Fields

  • Book Subtitle: École d'Été de Probabilités de Saint-Flour XXXVIII – 2008

  • Authors: Yves Le Jan

  • Series Title: Lecture Notes in Mathematics

  • DOI: https://doi.org/10.1007/978-3-642-21216-1

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: Springer-Verlag Berlin Heidelberg 2011

  • Softcover ISBN: 978-3-642-21215-4Published: 06 July 2011

  • eBook ISBN: 978-3-642-21216-1Published: 06 July 2011

  • Series ISSN: 0075-8434

  • Series E-ISSN: 1617-9692

  • Edition Number: 1

  • Number of Pages: VIII, 124

  • Number of Illustrations: 9 b/w illustrations

  • Topics: Probability Theory and Stochastic Processes, Potential Theory

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