Related papers: Lamplighter random walks on fractals
In this note, we try to analyze and clarify the intriguing interplay between some counting problems related to specific thermalized weighted graphs and random walks consistent with such graphs.
In this paper random walks on the Penrose lattice are investigated. Heat kernel estimates and the invariance principle are shown.
The relation between the expectation values computed in the random walk theory, and the heat kernel method for the diffusion equation is explained concretely. The random walk is also realized by simulations and their statistical…
In this article, local limit theorems for sequences of simple random walks on graphs are established. The results formulated are motivated by a variety of random graph models, and explanations are provided as to how they apply to…
For any graph having a suitable uniform Poincare inequality and volume growth regularity, we establish two-sided Gaussian transition density estimates and parabolic Harnack inequality, for constant speed continuous time random walks…
In this paper we consider a time-continuous random walk in $\mathbb{Z}^d$ in a dynamical random environment with symmetric jump rates to nearest neighbours. We assume that these random conductances are stationary and ergodic and, moreover,…
Consider the long-range percolation model on the integer lattice $\mathbb{Z}^d$ in which all nearest-neighbour edges are present and otherwise $x$ and $y$ are connected with probability $q_{x,y}:=1-\exp(-|x-y|^{-s})$, independently of the…
We consider time-dependent random walks among time-dependent conductances. For discrete time random walks, we show that, unlike the time-independent case, two-sided Gaussian heat kernel estimates are not stable under perturbations. This is…
The Martin boundary associated with the simple random walk on an example of partially oriented lattice is shown to be trivial by computing fine estimates of the Green kernel.
Graph-limit theory focuses on the convergence of sequences of graphs when the number of nodes becomes arbitrarily large. This framework defines a continuous version of graphs allowing for the study of dynamical systems on very large graphs,…
Consider a random walk $S_n=\sum_{i=1}^n X_i$ with independent and identically distributed real-valued increments with zero mean, finite variance and moment of order $2 + \delta$ for some $\delta>0$. For any starting point $x\in \mathbb R$,…
In this paper, we derive upper bounds for the heat kernel of the simple random walk on the infinite cluster of a supercritical long range percolation process. For any $d \geq 1$ and for any exponent $s \in (d, (d+2) \wedge 2d)$ giving the…
This paper presents necessary and sufficient conditions for on- and off-diagonal transition probability estimates for random walks on weighted graphs. On the integer lattice and on may fractal type graphs both the volume of a ball and the…
Kigami showed that a transient random walk on a deterministic infinite tree $T$ induces its trace process on the Martin boundary of $T$. In this paper, we will deal with trace processes on Martin boundaries of random trees instead of…
We study the trapping phenomenon of random walks in random environments of i.i.d. random conductances on the bonds of the grid $\mathbb{Z}^d$, the so-called random conductance model. Our main results concern the important model with…
We prove a Law of Iterated Logarithm for random walks on a family of diagonal products constructed by Brieussel and Zheng (2021). This provides a wide variety of new examples of Law of Iterated Logarithm behaviours for random walks on…
We consider the group of permutations of the vertices of a lattice. A random walk is generated by unit steps that each interchange two nearest neighbor vertices of the lattice. We study the heat equation on the permutation group, using the…
We consider learning on graphs, guided by kernels that encode similarity between vertices. Our focus is on random walk kernels, the analogues of squared exponential kernels in Euclidean spaces. We show that on large, locally treelike,…
We study discrete time random walks in an environment of i.i.d. non-negative bounded conductances in $\mathbb{Z}^d$. We are interested in the anomaly of the heat-kernel decay. We improve recent results and techniques.
Let a simple random walk run inside a torus of dimension three or higher for a number of steps which is a constant proportion of the volume. We examine geometric properties of the range, the random subgraph induced by the set of vertices…