Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 1903.07990

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

Kazuki Okamura
View original source ↗Read source document ↗Export metadata ↓

Abstract

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 Erd\"os (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.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Kazuki Okamura. 2019-03-19. Some results for range of random walk on graph with spectral dimension two. https://doi.org/10.1007/s10959-020-01013-0

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection

In this paper, we study averaging principles for nonautonomous multiscale stochastic Burgers equations with reflection. First, we derive a general averaging principle applicable to such equations under minimal assumptions. Subsequently, since the coefficients of the obtained averaged equation still depend on the small scaling parameter $\e$, we impose either periodic or asymptotic conditions on the coefficients, thereby obtain two distinct averaged equations whose coefficients are independent of $\e$ and establish two averaging principles. Stopping times and Khasminskii's time discretization schemes play an important role. Finally, a concrete example is provided to illustrate the applicability and validity of the theoretical results.

math.PR↗
arXiv2026

Spectral properties of Random Matrices

We give the theoretical foundations of random matrix theory through the definitions of a random matrix, a random probability measure and the corresponding empirical spectral distribution. The technical tool we use is the Stieltjes transform method through which we prove optimal convergence of the empirical spectral distribution of random sample covariance matrices to the deterministic Marchenko-Pastur distribution. We also give new results about the rigidity of the eigenvalues of this random sample covariance matrix and the rate of their convergence. We then define the Dyson equation method to prove new local laws about a random matrix model that interpolates between the Marchenko-Pastur distribution, the elliptical law and the circular law. Through our work these local laws can be considered universal.

math.PR↗
arXiv2026

Moments approach for the elephant random walk

We discuss the method of moments for the one-dimensional elephant random walk (ERW). We first derive a differential recurrence relation for the characteristic function of the ERW, which yields a corresponding system of recurrence relations for its moments. We then obtain asymptotic approximations for the moments in each of the three parameter regimes of the ERW. Finally, by establishing the convergence of the moments and verifying the corresponding moment-determinacy conditions, we identify the limiting distributions of the ERW in each regime.

math.PR↗