Related papers: Activated random walk on the comb
We study the time evolution of continuous-time quantum walks on randomly changing graphs. At certain moments edges of the graph appear or disappear with a given probability. We focus on the case when the time interval between subsequent…
Bootstrap percolation on a graph iteratively enlarges a set of occupied sites by adjoining points with at least $\theta$ occupied neighbors. The initially occupied set is random, given by a uniform product measure, and we say that spanning…
A fundamental problem on graph-structured data is that of quantifying similarity between graphs. Graph kernels are an established technique for such tasks; in particular, those based on random walks and return probabilities have proven to…
We introduce a new self-interacting random walk on the integers in a dynamic random environment and show that it converges to a pure diffusion in the scaling limit. We also find a lower bound on the diffusion coefficient in some special…
We study biased random walks on dynamical percolation on $\mathbb{Z}^d$. We establish a law of large numbers and an invariance principle for the random walk using regeneration times. Moreover, we verify that the Einstein relation holds, and…
We show how to compute the probabilities of various connection topologies for uniformly random spanning trees on graphs embedded in surfaces. As an application, we show how to compute the "intensity" of the loop-erased random walk in…
Techniques of `dynamic renormalization', developed earlier for undirected percolation and the contact model, are adapted to the setting of directed percolation, thereby obtaining solutions of several problems for directed percolation on…
Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on $\mathbb{Z}^2$ is $\mathrm{SLE}_2$. We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into…
We describe the critical window for percolation in the universality class of sparse growing random graphs. In our models, vertices arrive sequentially and connect independently to each earlier vertex $v$ with probability proportional to a…
There exists a Lipschitz embedding of a d-dimensional comb graph (consisting of infinitely many parallel copies of Z^{d-1} joined by a perpendicular copy) into the open set of site percolation on Z^d, whenever the parameter p is close…
Analyzing the mixing time of random walks is a well-studied problem with applications in random sampling and more recently in graph partitioning. In this work, we present new analysis of random walks and evolving sets using more…
We pose a new and intriguing question motivated by distributed computing regarding random walks on graphs: How long does it take for several independent random walks, starting from the same vertex, to cover an entire graph? We study the…
The cycle double cover conjecture is a long standing problem in graph theory, which links local properties, the valency of a vertex and no bridges, and a global property of the graph, being covered by a particular set of cycles. We prove…
We study continuous time quantum random walk on a comb with infinite teeth and show that the return probability to the starting point decays with time $t$ as $t^{-1}$. We analyse the diffusion along the spine and into the teeth and show…
We derive an exact closed-form analytical expression for the distribution of the cover time for a random walk over an arbitrary graph. In special case, we derive simplified exact expressions for the distributions of cover time for a…
Random walk on changing graphs is considered. For sequences of finite graphs increasing monotonically towards a limiting infinite graph, we establish transition probability upper bounds. It yields sufficient transience criteria for simple…
Dynamic graphs have emerged as an appropriate model to capture the changing nature of many modern networks, such as peer-to-peer overlays and mobile ad hoc networks. Most of the recent research on dynamic networks has only addressed the…
The problem of a restricted random walk on graphs which keeps track of the number of immediate reversal steps is considered by using a transfer matrix formulation. A closed-form expression is obtained for the generating function of the…
We consider the activated random walk model on general vertex-transitive graphs. A central question in this model is whether the critical density $\mu_c$ for sustained activity is strictly between 0 and 1. It was known that $\mu_c>0$ on…
We study the asymptotic behavior of the critical density of the activated random walk model as the sleep rate $\lambda$ tends to $0$ and $\infty$. For large $\lambda$, we prove new lower bounds in dimensions 1 and 2, showing that in one…