The mass of sites visited by a random walk on an infinite graph

Gibson, Lee R.

doi:10.1214/ejp.v13-531

Cited by 6 publications

(5 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There are many results for ranges on the integer lattices Z d ([DE, JO, JP70, JP71, JP72-1, JP72-2, JP74, S, DV, L, LR, H98, H01, HK01, HK02, BK, BR, BCR, LV]). On the other hand, there are relatively few results for ranges on more general (deterministic or random) infinite connected simple graphs ( [BIK,G,KM,KoN,O14,R]). Difficulties for graphs, especially for non vertex-transitive graphs, arise from the lack of spacial homogeneity of graphs.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Some results for range of random walk on graph with spectral dimension two

Okamura

2019

Preprint

View full text Add to dashboard Cite

We consider the range of the simple random walk on graphs with spectral dimension two. We give a form of strong law of large numbers under a certain uniform condition, which is satisfied by not only the square integer lattice but also a class of fractal graphs. Our results imply the strong law of large numbers on the square integer lattice established by Dvoretzky and Erdös (1951). Our proof does not depend on spacial homogeneity of space and gives a new proof of the strong law of large numbers on the lattice.We also show that the behavior of appropriately scaled expectations of the range is stable with respect to every "finite modification" of the two-dimensional integer lattice, and furthermore we construct a recurrent graph such that the uniform condition holds but the scaled expectations fluctuate. As an application, we establish a form of law of the iterated logarithms for lamplighter random walks in the case that the spectral dimension of the underlying graph is two.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

Some results for range of random walk on graph with spectral dimension two

Okamura

2019

Preprint

View full text Add to dashboard Cite

show abstract

“…The lower bound portion of Theorem 1 does not differ substantially from the classical approach [7], [14], requiring only the careful estimation of the probability that the sausage remains inside of a ball of radius r until at least time r β V (x, r). To treat the more challenging upper bound result, we turn to a coarse graining method out of the field of random media [5], [11], [22].…”

Section: Resultsmentioning

confidence: 99%

“…A result of this type appears in [12], and an exact discrete analogue to the below in [10]. Lemma 2.…”

Section: Upper Boundmentioning

confidence: 89%

“…Although the case of the translation invariant spaces Z d [7], R d [23], and Cayley graphs of finitely generated polynomial growth groups [8], [9], as well as the potentially translation invariant infinite graphs with Gaussian and sub-Gaussian estimates [11] have been studied, the setting of a local Dirichlet space broadens the class of examples that one can consider to a large class of continuous spaces without explicit symmetries. (2)).…”

Section: Examplesmentioning

confidence: 99%

“…Since the proofs related to the known behavior of the Wiener sausage in R d rely on the translation and scale invariance of R d and Brownian motion, the techniques of those proofs cannot be applied directly in the local Dirichlet space context. Instead, we will follow Date: 28 July, 2010. the solution technique of the discrete analogue of our problem (random walk on a nontransitive graph) appearing in [11]. In this process, we will make use of the relationship between the exponential moments of the volume of the sausage and the annealed survival probability of the diffusion process among Poisson distributed hard obstacles.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Rate of Decay of the Wiener Sausage in Local Dirichlet Space

Gibson

Pivarski

2014

J Theor Probab

View full text Add to dashboard Cite

In the context of a heat kernel diffusion which admits a Gaussian type estimate with parameter β on a local Dirichlet space, we consider the log asymptotic behavior of the negative exponential moments of the Wiener sausage. We show that the log asymptotic behavior up to time t β V (x,t) is V (x,t), which is analogous to the Euclidean result.Here V (x,t) represents the mass of the ball of radius t about a point x of the local Dirichlet space. The proof uses a known coarse graining technique to obtain the upper asymptotic, but must be adapted for use without translation invariance in this setting. This result provides the first such asymptotics for several other contexts, including diffusions on complete Riemannian manifolds with non-negative Ricci curvature.

show abstract