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Random walks on disordered media and their scaling limits : École d'Été de Probabilités de Saint-Flour XL-2010 / Takashi Kumagai.

By: Material type: TextTextSeries: Lecture notes in mathematics (Springer-Verlag) ; 2101.Publisher: Cham [Switzerland] ; New York : Springer, [2014]Copyright date: ©2014Description: 1 online resource (x, 147 pages) : illustrationsContent type:
  • text
Media type:
  • computer
Carrier type:
  • online resource
ISBN:
  • 9783319031521
  • 331903152X
Other title:
  • École d'Été de Probabilités de Saint-Flour XL-2010
Subject(s): Additional physical formats: Print version:: Random walks on disordered media and their scaling limits.DDC classification:
  • 519.2 23
LOC classification:
  • QA274.73
Online resources:
Contents:
Introduction -- Weighted graphs and the associated Markov chains -- Heat kernel estimates -- General theory -- Heat kernel estimates using effective resistance -- Heat kernel estimates for random weighted graphs -- Alexander-Orbach conjecture holds when two-point functions behave nicely -- Further results for random walk on IIC -- Random conductance model.
Summary: In these lecture notes, we will analyze the behavior of random walk on disordered mediaby means ofboth probabilistic and analytic methods, and will study the scalinglimits. We will focus on the discrete potential theory and how the theory is effectively used in the analysis of disordered media. Thefirst few chapters of the notes can be used as an introduction to discrete potential theory. Recently, there has beensignificantprogress on thetheoryof random walkon 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 showedtheanomalous 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.
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Includes bibliographical references (pages 135-143) and index.

Introduction -- Weighted graphs and the associated Markov chains -- Heat kernel estimates -- General theory -- Heat kernel estimates using effective resistance -- Heat kernel estimates for random weighted graphs -- Alexander-Orbach conjecture holds when two-point functions behave nicely -- Further results for random walk on IIC -- Random conductance model.

In these lecture notes, we will analyze the behavior of random walk on disordered mediaby means ofboth probabilistic and analytic methods, and will study the scalinglimits. We will focus on the discrete potential theory and how the theory is effectively used in the analysis of disordered media. Thefirst few chapters of the notes can be used as an introduction to discrete potential theory. Recently, there has beensignificantprogress on thetheoryof random walkon 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 showedtheanomalous 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.

English.

Online resource; title from PDF title page (SpringerLink, viewed February 25, 2014).

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