Authors:
- Starts from basics on discrete potential theory
- Contains many interesting examples of disordered media with anomalous heat conduction
- Anomalous behavior of random walk at criticality on random media
- Contains recent developments on random conductance models
- Includes supplementary material: sn.pub/extras
Part of the book series: Lecture Notes in Mathematics (LNM, volume 2101)
Part of the book sub series: École d'Été de Probabilités de Saint-Flour (LNMECOLE)
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Table of contents (8 chapters)
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Front Matter
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Back Matter
About this book
In these lecture notes, we will analyze the behavior of random walk on disordered media by means of both probabilistic and analytic methods, and will study the scaling limits. We will focus on the discrete potential theory and how the theory is effectively used in the analysis of disordered media. The first few chapters of the notes can be used as an introduction to discrete potential theory.
Recently, there has been significant progress on the theory of random walk on disordered media such as fractals and random media. Random walk on a percolation cluster(‘the ant in the labyrinth’)is one of the typical examples. In 1986, H. Kesten showed the anomalous behavior of a random walk on a percolation cluster at critical probability. Partly motivated by this work, analysis and diffusion processes on fractals have been developed since the late eighties. As a result, various new methods have been produced to estimate heat kernels on disordered media. These developments are summarized in the notes.
Authors and Affiliations
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Research Institute for Mathematical Scie, Kyoto University, Kyoto, Japan
Takashi Kumagai
Bibliographic Information
Book Title: Random Walks on Disordered Media and their Scaling Limits
Book Subtitle: École d'Été de Probabilités de Saint-Flour XL - 2010
Authors: Takashi Kumagai
Series Title: Lecture Notes in Mathematics
DOI: https://doi.org/10.1007/978-3-319-03152-1
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer International Publishing Switzerland 2014
Softcover ISBN: 978-3-319-03151-4Published: 04 February 2014
eBook ISBN: 978-3-319-03152-1Published: 25 January 2014
Series ISSN: 0075-8434
Series E-ISSN: 1617-9692
Edition Number: 1
Number of Pages: X, 147
Number of Illustrations: 5 b/w illustrations
Topics: Probability Theory and Stochastic Processes, Mathematical Physics, Potential Theory, Discrete Mathematics