Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree
نویسندگان
چکیده
The first main result of this paper is that the law of the (rescaled) two-dimensional uniform spanning tree is tight in a space whose elements are measured, rooted real trees continuously embedded into Euclidean space. Various properties of the intrinsic metrics, measures and embeddings of the subsequential limits in this space are obtained, with it being proved in particular that the Hausdor↵ dimension of any limit in its intrinsic metric is almost surely equal to 8/5. In addition, the tightness result is applied to deduce that the annealed law of the simple random walk on the two-dimensional uniform spanning tree is tight under a suitable rescaling. For the limiting processes, which are di↵usions on random real trees embedded into Euclidean space, detailed transition density estimates are derived.
منابع مشابه
Scaling limits of loop - erased random walks and uniform spanning trees
The uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of these su...
متن کاملThe loop - erased random walk and the uniform spanning tree on the four - dimensional discrete torus
Let x and y be points chosen uniformly at random from Z4n, the four-dimensional discrete torus with side length n. We show that the length of the loop-erased random walk from x to y is of order n(logn), resolving a conjecture of Benjamini and Kozma. We also show that the scaling limit of the uniform spanning tree on Z4n is the Brownian continuum random tree of Aldous. Our proofs use the techniq...
متن کاملScaling Limits of the Uniform Spanning Tree and Loop-erased Random Walk on Finite Graphs
Let x and y be chosen uniformly in a graph G. We find the limiting distribution of the length of a loop-erased random walk from x to y on a large class of graphs that include the torus Zn for d ≥ 5. Moreover, on this family of graphs we show that a suitably normalized finite-dimensional scaling limit of the uniform spanning tree is a Brownian continuum random tree.
متن کاملSpectral dimension and random walks on the two dimensional uniform spanning tree
We study simple random walk on the uniform spanning tree on Z2. We obtain estimates for the transition probabilities of the random walk, the distance of the walk from its starting point after n steps, and exit times of both Euclidean balls and balls in the intrinsic graph metric. In particular, we prove that the spectral dimension of the uniform spanning tree on Z2 is 16/13 almost surely.
متن کاملInfinite volume limits of high-dimensional sandpile models
We study the Abelian sandpile model on Z. In d ≥ 3 we prove existence of the infinite volume addition operator, almost surely with respect to the infinite volume limit μ of the uniform measures on recurrent configurations. We prove the existence of a Markov process with stationary measure μ, and study ergodic properties of this process. The main techniques we use are a connection between the st...
متن کامل