5,657 research outputs found

    Transience and recurrence of random walks on percolation clusters in an ultrametric space

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    We study existence of percolation in the hierarchical group of order NN, which is an ultrametric space, and transience and recurrence of random walks on the percolation clusters. The connection probability on the hierarchical group for two points separated by distance kk is of the form ck/Nk(1+δ),δ>1c_k/N^{k(1+\delta)}, \delta>-1, with ck=C0+C1logk+C2kαc_k=C_0+C_1\log k+C_2k^\alpha, non-negative constants C0,C1,C2C_0, C_1, C_2, and α>0\alpha>0. Percolation was proved in Dawson and Gorostiza (2013) for δ0\delta0, with α>2\alpha>2. In this paper we improve the result for the critical case by showing percolation for α>0\alpha>0. We use a renormalization method of the type in the previous paper in a new way which is more intrinsic to the model. The proof involves ultrametric random graphs (described in the Introduction). The results for simple (nearest neighbour) random walks on the percolation clusters are: in the case δ<1\delta<1 the walk is transient, and in the critical case δ=1,C2>0,α>0\delta=1, C_2>0,\alpha>0, there exists a critical αc(0,)\alpha_c\in(0,\infty) such that the walk is recurrent for α<αc\alpha<\alpha_c and transient for α>αc\alpha>\alpha_c. The proofs involve graph diameters, path lengths, and electric circuit theory. Some comparisons are made with behaviours of random walks on long-range percolation clusters in the one-dimensional Euclidean lattice.Comment: 27 page

    Surprise probabilities in Markov chains

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    In a Markov chain started at a state xx, the hitting time τ(y)\tau(y) is the first time that the chain reaches another state yy. We study the probability Px(τ(y)=t)\mathbf{P}_x(\tau(y) = t) that the first visit to yy occurs precisely at a given time tt. Informally speaking, the event that a new state is visited at a large time tt may be considered a "surprise". We prove the following three bounds: 1) In any Markov chain with nn states, Px(τ(y)=t)nt\mathbf{P}_x(\tau(y) = t) \le \frac{n}{t}. 2) In a reversible chain with nn states, Px(τ(y)=t)2nt\mathbf{P}_x(\tau(y) = t) \le \frac{\sqrt{2n}}{t} for t4n+4t \ge 4n + 4. 3) For random walk on a simple graph with n2n \ge 2 vertices, Px(τ(y)=t)4elognt\mathbf{P}_x(\tau(y) = t) \le \frac{4e \log n}{t}. We construct examples showing that these bounds are close to optimal. The main feature of our bounds is that they require very little knowledge of the structure of the Markov chain. To prove the bound for random walk on graphs, we establish the following estimate conjectured by Aldous, Ding and Oveis-Gharan (private communication): For random walk on an nn-vertex graph, for every initial vertex xx, \[ \sum_y \left( \sup_{t \ge 0} p^t(x, y) \right) = O(\log n). \

    Contact and voter processes on the infinite percolation cluster as models of host-symbiont interactions

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    We introduce spatially explicit stochastic processes to model multispecies host-symbiont interactions. The host environment is static, modeled by the infinite percolation cluster of site percolation. Symbionts evolve on the infinite cluster through contact or voter type interactions, where each host may be infected by a colony of symbionts. In the presence of a single symbiont species, the condition for invasion as a function of the density of the habitat of hosts and the maximal size of the colonies is investigated in details. In the presence of multiple symbiont species, it is proved that the community of symbionts clusters in two dimensions whereas symbiont species may coexist in higher dimensions.Comment: Published in at http://dx.doi.org/10.1214/10-AAP734 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Giant vacant component left by a random walk in a random d-regular graph

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    We study the trajectory of a simple random walk on a d-regular graph with d>2 and locally tree-like structure as the number n of vertices grows. Examples of such graphs include random d-regular graphs and large girth expanders. For these graphs, we investigate percolative properties of the set of vertices not visited by the walk until time un, where u>0 is a fixed positive parameter. We show that this so-called vacant set exhibits a phase transition in u in the following sense: there exists an explicitly computable threshold u* such that, with high probability as n grows, if u<u*, then the largest component of the vacant set has a volume of order n, and if u>u*, then it has a volume of order log(n). The critical value u* coincides with the critical intensity of a random interlacement process (introduced by Sznitman [arXiv:0704.2560]) on a d-regular tree. We also show that the random interlacement model describes the structure of the vacant set in local neighbourhoods

    Universality of trap models in the ergodic time scale

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    Consider a sequence of possibly random graphs GN=(VN,EN)G_N=(V_N, E_N), N1N\ge 1, whose vertices's have i.i.d. weights {WxN:xVN}\{W^N_x : x\in V_N\} with a distribution belonging to the basin of attraction of an α\alpha-stable law, 0<α<10<\alpha<1. Let XtNX^N_t, t0t \ge 0, be a continuous time simple random walk on GNG_N which waits a \emph{mean} WxNW^N_x exponential time at each vertex xx. Under considerably general hypotheses, we prove that in the ergodic time scale this trap model converges in an appropriate topology to a KK-process. We apply this result to a class of graphs which includes the hypercube, the dd-dimensional torus, d2d\ge 2, random dd-regular graphs and the largest component of super-critical Erd\"os-R\'enyi random graphs
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