Random Walks on Sierpiński Graphs: Hyperbolicity and Stochastic Homogenization
نویسنده
چکیده
We introduce two new techniques to the analysis on fractals. One is based on the presentation of the fractal as the boundary of a countable Gromov hyperbolic graph, whereas the other one consists in taking all possible “backward” extensions of the above hyperbolic graph and considering them as the classes of a discrete equivalence relation on an appropriate compact space. Illustrating these techniques on the example of the Sierpiński gasket (the associated hyperbolic graph is called the Sierpiński graph), we show that the Sierpiński gasket can be identified with the Martin and the Poisson boundaries for fairly general classes of Markov chains on the Sierpiński graph.
منابع مشابه
A PRELUDE TO THE THEORY OF RANDOM WALKS IN RANDOM ENVIRONMENTS
A random walk on a lattice is one of the most fundamental models in probability theory. When the random walk is inhomogenous and its inhomogeniety comes from an ergodic stationary process, the walk is called a random walk in a random environment (RWRE). The basic questions such as the law of large numbers (LLN), the central limit theorem (CLT), and the large deviation principle (LDP) are ...
متن کاملUniform spanning trees on Sierpiński graphs
We study spanning trees on Sierpiński graphs (i.e., finite approximations to the Sierpiński gasket) that are chosen uniformly at random. We construct a joint probability space for uniform spanning trees on every finite Sierpiński graph and show that this construction gives rise to a multi-type Galton-Watson tree. We derive a number of structural results, for instance on the degree distribution....
متن کاملh . PR ] 1 5 Ja n 20 02 On the physical relevance of random walks : an example of random walks on a randomly oriented lattice ∗
Random walks on general graphs play an important role in the understanding of the general theory of stochastic processes. Beyond their fundamental interest in probability theory, they arise also as simple models of physical systems. A brief survey of the physical relevance of the notion of random walk on both undirected and directed graphs is given followed by the exposition of some recent resu...
متن کاملCoalescent Random Walks on Graphs
Inspired by coalescent theory in biology, we introduce a stochastic model called ”multi-person simple random walks” or “coalescent random walks” on a graph G. There are any finite number of persons distributed randomly at the vertices of G. In each step of this discrete time Markov chain, we randomly pick up a person and move it to a random adjacent vertex. To study this model, we introduce the...
متن کاملOn coalescence time in graphs-When is coalescing as fast as meeting?
Coalescing random walks is a fundamental stochastic process, where a set of particles perform independent discrete-time random walks on an undirected graph. Whenever two or more particles meet at a given node, they merge and continue as a single random walk. The coalescence time is defined as the expected time until only one particle remains, starting from one particle at every node. Despite re...
متن کامل