TY - BOOK
AU - Kumagai, Takashi.
ED - SpringerLink (Online service)
TI - Random Walks on Disordered Media and their Scaling Limits: École d'Été de Probabilités de Saint-Flour XL - 2010
T2 - Lecture Notes in Mathematics,
SN - 9783319031
AV - QA273.A1-274.9
U1 - 519.2 23
PY - 2014///
CY - Cham
PB - Springer International Publishing, Imprint: Springer
KW - Mathematics
KW - Potential theory (Mathematics)
KW - Distribution (Probability theory)
KW - Probability Theory and Stochastic Processes
KW - Mathematical Physics
KW - Potential Theory
KW - Discrete Mathematics
N1 - Introduction -- Weighted graphs and the associated Markov chains -- Heat kernel estimates â€“ 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
N2 - In these lecture notes, we will analyze the behavior of random walk on disordered mediaÂ by means ofÂ both probabilistic and analytic methods, and will study the scalingÂ limits. We will focus on the discrete potential theory and how the theory is effectively used in the analysis of disordered media.Â TheÂ first few chapters of the notes can be used as an introduction to discrete potential theory. Â Recently, there has beenÂ significantÂ progress on theÂ theoryÂ of random walkÂ on disordered media such as fractals and random media.Â Random walk on a percolation clusterÂ (â€˜the ant in the labyrinthâ€™)Â is one of the typical examples. In 1986, H. Kesten showedÂ theÂ 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
UR - http://dx.doi.org/10.1007/978-3-319-03152-1
ER -