منابع مشابه
Counting Planar Random Walk Holes
We study two variants of the notion of holes formed by planar simple random walk of time duration 2n and the areas associated with them. We prove in both cases that the number of holes of area greater than A(n), where {A(n)} is an increasing sequence, is, up to a logarithmic correction term, asymptotic to n · A(n) for a range of large holes, thus confirming an observation by Mandelbrot. A conse...
متن کاملCounting Hamiltonian cycles on planar random lattices
Hamiltonian cycles on planar random lattices are considered. The generating function for the number of Hamiltonian cycles is obtained and its singularity is studied. Hamiltonian cycles have often been used to model collapsed polymer globules[1]. A Hamiltonian cycle of a graph is a closed path which visits each of the vertices once and only once. The number of Hamiltonian cycles on a lattice cor...
متن کاملThe growth exponent for planar loop-erased random walk
We give a new proof of a result of Kenyon that the growth exponent for loop-erased random walks in two dimensions is 5/4. The proof uses the convergence of LERW to Schramm-Loewner evolution with parameter 2, and is valid for irreducible bounded symmetric random walks on any discrete lattice of R2.
متن کاملLoop-Erased Random Walk and Poisson Kernel on Planar Graphs
Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on Z2 is SLE2. We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into C so that edges do not cross one another). We show that if the scaling limit of the random walk is planar Brownian motion, then the scaling limit of its loop-erasure is SLE2. Our main co...
متن کاملSome Estimates for Planar Random Walk and Brownian Motion
The purpose of this note is to collect in one place a few results about simple random walk and Brownian motion which are often useful. These include standard results such as Beurling estimates, large deviation estimates, and a method for coupling the two processes, as well as solutions to the discrete Dirichlet problem in various domains which, to the author’s knowledge, have not been published...
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ژورنال
عنوان ژورنال: The Annals of Probability
سال: 2008
ISSN: 0091-1798
DOI: 10.1214/009117907000000204