Related papers: Cycle density in infinite Ramanujan graphs
We give new general formulas for the asymptotics of the number of spanning trees of a large graph. A special case answers a question of McKay (1983) for regular graphs. The general answer involves a quantity for infinite graphs that we call…
Recently Lubetzky and Peres showed that simple random walks on a sequence of $d$-regular Ramanujan graphs $G_n=(V_n,E_n)$ of increasing sizes exhibit cutoff in total variation around the diameter lower bound $\frac{d}{d-2}\log_{d-1}|V_n| $.…
We study the behavior of Random Walk in Random Environment (RWRE) on trees in the critical case left open in previous work. Representing the random walk by an electrical network, we assume that the ratios of resistances of neighboring edges…
It has been recently proposed that the natural connectivity can be used to characterize efficiently the robustness of complex networks. The natural connectivity quantifies the redundancy of alternative routes in the network by evaluating…
We construct a bounded degree graph $G$, such that a simple random walk on it is transient but the random walk path (i.e., the subgraph of all the edges the random walk has crossed) has only finitely many cutpoints, almost surely. We also…
Non-trivial linear bounds are obtained for the displacement of a random walk in a dynamic random environment given by a one-dimensional simple symmetric exclusion process in equilibrium. The proof uses an adaptation of multiscale…
We study the entropy of the distribution of the set R_n of vertices visited by a simple random walk on a graph with bounded degrees in its first n steps. It is shown that this quantity grows linearly in the expected size of R_n if the graph…
A random walk is a basic stochastic process on graphs and a key primitive in the design of distributed algorithms. One of the most important features of random walks is that, under mild conditions, they converge to a stationary distribution…
The total-variation cutoff phenomenon has been conjectured to hold for simple random walk on all transitive expanders. However, very little is actually known regarding this conjecture, and cutoff on sparse graphs in general. In this paper…
We consider a class of biased random walks on infinite graphs and present several general results on the spectral radius of biased random walk.
We study the properties of random walks on complex trees. We observe that the absence of loops reflects in physical observables showing large differences with respect to their looped counterparts. First, both the vertex discovery rate and…
In this paper we show that on bounded degree graphs and general trees, the cover time of the simple random walk is asymptotically equal to the product of the number of edges and the square of the expected supremum of the Gaussian free field…
We initiate the study of property testing in arbitrary planar graphs. We prove that bipartiteness can be tested in constant time, improving on the previous bound of $\tilde{O}(\sqrt{n})$ for graphs on $n$ vertices. The constant-time…
Recently, random walks on dynamic graphs have been studied because of their adaptivity to the time-varying structure of real-world networks. In general, there is a tremendous gap between static and dynamic graph settings for the lazy simple…
We focus on recurrent random walks in random environment (RWRE) on Galton-Watson trees. The range of these walks, that is the number of sites visited at some fixed time, has been studied in three different papers [AC18], [AdR17] and [dR16].…
We study the minimal spanning arborescence which is the directed analogue of the minimal spanning tree, with a particular focus on its infinite volume limit and its geometric properties. We prove that in a certain large class of transient…
In this work, we are interested in the set of visited vertices of a tree $\mathbb{T}$ by a randomly biased random walk $\mathbb{X}:=(X_n,n \in \mathbb{N})$. The aim is to study a generalized range, that is to say the volume of the trace of…
We show that the transience or recurrence of a random walk in certain random environments on an arbitrary infinite locally finite tree is determined by the branching number of the tree, which is a measure of the average number of branches…
Classical random walks on well-behaved graphs are rapidly mixing towards the uniform distribution. Moore and Russell showed that a continuous quantum walk on the hypercube is instantaneously uniform mixing. We show that the continuous-time…
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…