相关论文: Return probability for the loop-erased random walk…
The height probabilities for the recurrent configurations in the Abelian Sandpile Model on the square lattice have analytic expressions, in terms of multidimensional quadratures. At first, these quantities have been evaluated numerically…
For the Abelian sandpile model on Sierpinski graphs, we investigate several statistics such as average height, height probabilities and looping constant. In particular, we calculate the expected average height of a recurrent sandpile on the…
We show that the dissipative Abelian sandpile on a graph L can be related to a random walk on a graph which consists of L extended with a trapping site. From this relation it can be shown, using exact results and a scaling assumption, that…
We prove an estimate for the probability that a simple random walk in a simply connected subset A of Z^2 starting on the boundary exits A at another specified boundary point. The estimates are uniform over all domains of a given inradius.…
We consider a continuous height version of the Abelian sandpile model with small amount of bulk dissipation gamma > 0 on each toppling, in dimensions d = 2, 3. In the limit gamma -> 0, we give a power law upper bound, based on coupling, on…
We give an asymptotic formula for the single site height distribution of Abelian sandpiles on $\mathbb{Z}^d$ as $d \to \infty$, in terms of $\mathsf{Poisson}(1)$ probabilities. We provide error estimates.
We have studied the probability distribution of the perimeter and the area of the k-th largest erased-loop in loop-erased random walks in two-dimensions for k = 1 to 3. For a random walk of N steps, for large N, the average value of the…
We show that in the loop-erased random walk problem, the exponent characterizing probability distribution of areas of erased loops is superuniversal. In d-dimensions, the probability that the erased loop has an area A varies as A^{-2} for…
We give a simple formula for the looping rate of loop-erased random walk on a finite planar graph. The looping rate is closely related to the expected amount of sand in a recurrent sandpile on the graph. The looping rate formula is…
In this article, we consider several models of random walks in one or several dimensions, additionally allowing, at any unit of time, a reset (or "catastrophe") of the walk with probability $q$. We establish the distribution of the final…
We compute the lattice 1-site probabilities, on the upper half-plane, of the four height variables in the two-dimensional Abelian sandpile model. We find their exact scaling form when the insertion point is far from the boundary, and when…
A return probability of random walks is one of the interesting subjects. As it is well known, the return probability strongly depends on the structure of the space where the random waker moves. On the other hand, the return probability of…
The paper develops one-parametric family of the sand-piles dealing with the grains' local losses on the fixed amount. The family exhibits the crossover between the models with deterministic and stochastic relaxation. The mean height of the…
An Abelian sandpile model is considered on the Husimi lattice of triangles with an arbitrary coordination number q. Exact expressions for the distribution of height probabilities in the Self-Organized Critical state are derived.
Motivated by the dissipative abelian sandpile model, we analyze the trajectories of a one-dimensional random walk in a landscape of soft traps. These traps, placed at increasing distances from each other, correspond to dissipative sites in…
We study analytically a simple random walk model on a one-dimensional lattice, where at each time step the walker resets to the maximum of the already visited positions (to the rightmost visited site) with a probability $r$, and with…
We consider a discrete random walk on a diagonal lattice in two and three dimensions and obtain explicit solutions of absorption probabilities and probabilities of return in several domains. In three dimensions we consider both the cube and…
We study the excited random walk, in which a walk that is at a site that contains cookies eats one cookie and then hops to the right with probability p and to the left with probability q=1-p. If the walk hops onto an empty site, there is no…
The loop-erased random walk (LERW) in $\mathbb{Z}^4$ is the process obtained by erasing loops chronologically for simple random walk. We prove that the escape probability of the LERW renormalized by $(\log n)^{\frac{1}{3}}$ converges almost…
We consider the biased random walk on a tree constructed from the set of finite self-avoiding walks on a lattice, and use it to construct probability measures on infinite self-avoiding walks. The limit measure (if it exists) obtained when…