We consider the following dynamic Boolean model introduced by van den Berg, Meester and White (1997). At time 0, let the nodes of the graph be a Poisson point process in R d with constant intensity and let each node move independently according to Brownian motion. At any time t, we put an edge between every pair of nodes whose distance is at most r. We study three fundamental problems in this model: detection (the time until a target point-fixed or moving-is within distance r of some node of the graph); coverage (the time until all points inside a finite box are detected by the graph); and percolation (the time until a given node belongs to the infinite connected component of the graph). We obtain precise asymptotics for these quantities by combining ideas from stochastic geometry, coupling and multi-scale analysis.
We consider irreducible reversible discrete time Markov chains on a finite state space. Mixing times and hitting times are fundamental parameters of the chain. We relate them by showing that the mixing time of the lazy chain is equivalent to the maximum over initial states x and large sets A of the hitting time of A starting from x. We also prove that the first time when averaging over two consecutive time steps is close to stationarity is equivalent to the mixing time of the lazy version of the chain.
We study the capacity of the range of a transient simple random walk on Z d . Our main result is a central limit theorem for the capacity of the range for d ≥ 6. We present a few open questions in lower dimensions.
We study the scaling limit of the capacity of the range of a simple random walk on the integer lattice in dimension four. We establish a strong law of large numbers and a central limit theorem with a non-gaussian limit. The asymptotic behaviour is analogous to that found by Le Gall in '86 [28] for the volume of the range in dimension two.One easily passes from one representation in (1.1) to the other using the last passage decomposition formula, see (2.8) below. Denote by {S(n), n ∈ N} a simple random walk in Z 4 . For two integers m, n, the range R[m, n] (or simply R n when m = 0) in the time period [m, n] is defined as R[m, n] = {S(m), . . . , S(n)}.Our first result is a strong law of large numbers for Cap (R n ).Recently, van den Berg, Bolthausen and den Hollander [38] advocated a new geometric characteristic, the torsional rigidity of the complement of a Wiener sausage, as a way to probe the shape of the sausage. In order to obtain leading asymptotics for the torsional rigidity, one needs a law of large numbers for the capacity of a Wiener sausage, which is not proved yet in dimension four; see however our companion paper [5] for a partial result in this direction. Our Theorem 1.1 establishes these asymptotics for the discrete model, and thus prepares the study of torsional rigidity for random walk.
A recurrent graph G has the infinite collision property if two independent random walks on G, started at the same point, collide infinitely often a.s. We give a simple criterion in terms of Green functions for a graph to have this property, and use it to prove that a critical Galton-Watson tree with finite variance conditioned to survive, the incipient infinite cluster in Z d with d ≥ 19 and the uniform spanning tree in Z 2 all have the infinite collision property. For power-law combs and spherically symmetric trees, we determine precisely the phase boundary for the infinite collision property.We begin by defining what we mean by the finite/infinite collision property. Throughout this paper we will only consider connected graphs.Definition 1.1. Let G be a graph, and X, Y be independent (discrete time) simple random walks on G. We write P a,b for the law of the processbe the total number of collisions between X and Y . Ifthen G has the finite collision property. Ifthen G has the infinite collision property.We will see below that these are the only two possibilities.We recall the definition of Comb(Z): Definition 1.2. Comb(Z) is the graph with vertex set Z × Z and edge set {[(x, n), (x, m)] : |m − n| = 1} ∪ {[(x, 0), (y, 0)] : |x − y| = 1]} Definition 1.3. Following [10], we define the wedge comb with profile f , denoted Comb(Z, f ) to be the subgraph of Comb(Z) with vertex set V = {(x, y) ∈ Z 2 : 0 ≤ y ≤ f (x)} and edge set the set of edges of Comb(Z) with vertices in V . We write Comb(Z, α) for the wedge comb with profile f (k) = k α .In [10] it is proved that Comb(Z, α) has the infinite collision property when α < 1/5.We have the following phase transition:Theorem 1.4. (a) If α ≤ 1, then Comb(Z, α) has the infinite collision property.(b) If α > 1, then Comb(Z, α) has the finite collision property.We remark that the proofs of both (a) and (b) extend to the profiles of the form f (x) = C|x| α . Part (b) for 1 < α < 2 was also obtained independently by J. Beltran, D.Y. Chen, T. Mountford and D. Valesin (private communication).Remark 1.5. This theorem shows that if the 'teeth' in the comb are large then the finite collision property will hold, while it fails if they are small. However, there is no simple monotonicity property for the finite collision property: Comb(Z) has the finite collision property but is a subgraph of Z 2 , which does not.Further, we do not have any kind of 'bracketing' property for collisions: we have Comb(Z, 1) ⊂ Comb(Z, 2) ⊂ Z 2 ⊂ Comb(Z 2 ); and of these Comb(Z, 1) and Z 2 have the infinite collision property while the other two graphs have the finite collision property. (See [17] for the definition of Comb(Z 2 ), and the proof that it has the finite collision property.)
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