相关论文: Cutpoints and resistance of random walk paths
We prove that a simple random walk on quasi-transitive graphs with the volume growth being faster than any polynomial of degree 4 has a.s. infinitely many cut times, and hence infinitely many cutpoints. This confirms a conjecture raised by…
We give an example of a transient reversible Markov chain that almost surely has only a finite number of cutpoints. We explain how this is relevant to a conjecture of Diaconis and Freedman and a question of Kaimanovich. We also answer…
We consider transient nearest neighbor random walks on the positive part of the real line. We give criteria for the finiteness of the number of cutpoints and strong cutpoints. Examples and open problems are presented.
We prove that if $(X_n)_{n\geq 0}$ is a random walk on a transient graph such that the Green's function decays at least polynomially along the random walk, then $(X_n)_{n\geq 0}$ has infinitely many cut times almost surely. This condition…
Consider a transient near-critical (1,2) random walk on the positive half line. We give a criteria for the finiteness of the number of the skipped points (the points never visited) by the random walk. This result generalizes (partially) the…
A finite ergodic Markov chain exhibits cutoff if its distance to equilibrium remains close to its initial value over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Originally discovered in the…
In this paper, we study (1,2) and (2,1) random walks in varying environments on the lattice of positive half line. We assume that the transition probabilities at site $n$ are asymptotically constants as $n\rightarrow\infty.$ For (1,2)…
A constructive proof is given to the fact that any ergodic Markov chain can be realized as a random walk subject to a synchronizing road coloring. Redundancy (ratio of extra entropy) in such a realization is also studied.
We introduce the notion of a "random basic walk" on an infinite graph, give numerous examples, list potential applications, and provide detailed comparisons between the random basic walk and existing generalizations of simple random walks.…
We give conditions under which near-critical stochastic processes on the half-line have infinitely many or finitely many cutpoints, generalizing existing results on nearest-neighbour random walks to adapted processes with bounded increments…
A survey is presented of known results concerning simple random walk on the class of distance-regular graphs. One of the highlights is that electric resistance and hitting times between points can be explicitly calculated and given strong…
It is shown that transient graphs for the simple random walk do not admit a nearest neighbor transient Markov chain (not necessarily a reversible one), that crosses all edges with positive probability, while there is such chain for the…
A finite ergodic Markov chain is said to exhibit cutoff if its distance to stationarity remains close to 1 over a certain number of iterations and then abruptly drops to near 0 on a much shorter time scale. Discovered in the context of card…
Directed covers of finite graphs are also known as periodic trees or trees with finitely many cone types. We expand the existing theory of directed covers of finite graphs to those of infinite graphs. While the lower growth rate still…
We prove that metric graph with the minimal growth of the number of possible endpoints of a random walk is the union of several linear paths coming out of the same vertex
Refining previous results, we establish a sharp asymptotic estimate on the expected graph distance between the origin and the terminal point of the trace of the first $n$ steps of the walk. A similar conclusion is drawn for the resistance…
It is shown that the path of a simple random walk on any graph, consisting of all vertices visited and edges crossed by the walk, is almost surely a recurrent subgraph.
Let $P$ be the transition matrix of a finite, irreducible and reversible Markov chain. We say the continuous time Markov chain $X$ has transition matrix $P$ and speed $\lambda$ if it jumps at rate $\lambda$ according to the matrix $P$. Fix…
We investigate the mixing properties of a model of reversible Markov chains in random environment, which notably contains the simple random walk on the superposition of a deterministic graph and a second graph whose vertex set has been…
Markov chains are convenient means of generating realizations of networks with a given (joint or otherwise) degree distribution, since they simply require a procedure for rewiring edges. The major challenge is to find the right number of…