© 2014

Random Walks on Disordered Media and their Scaling Limits

École d'Été de Probabilités de Saint-Flour XL - 2010


Part of the Lecture Notes in Mathematics book series (LNM, volume 2101)

Also part of the École d'Été de Probabilités de Saint-Flour book sub series (LNMECOLE, volume 2101)

Table of contents

  1. Front Matter
    Pages i-x
  2. Takashi Kumagai
    Pages 1-2
  3. Takashi Kumagai
    Pages 21-41
  4. Takashi Kumagai
    Pages 79-93
  5. Takashi Kumagai
    Pages 95-134
  6. Back Matter
    Pages 135-150

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


60-xx,35K05,05C81,82B43,80M40 Heat kernel estimates Homogenization Markov chain Percolation Random media

Authors and affiliations

  1. 1.Research Institute for Mathematical ScieKyoto UniversityKyotoJapan

Bibliographic information

Industry Sectors
IT & Software
Finance, Business & Banking
Energy, Utilities & Environment
Oil, Gas & Geosciences