Consider an independent site percolation model on Z , with parameter p ∈ (0, 1), where all long-range connections in the axis directions are allowed. In this work we show that, given any parameter p, there exists an integer K(p) such that all binary sequences (words) ξ ∈ {0, 1}N can be seen simultaneously, almost surely, even if all connections with length larger than K(p) are suppressed. We al...

An infinite graph G has the property that a random walk in random environment on G defined by i.i.d. resistances with any common distribution is almost surely transient, if and only if for some p < 1, simple random walk is transient on a percolation cluster of G under bond percolation with parameter p.

Using percolation theory, we derive a conceptual definition of deconfinement in terms of cluster formation. The result is readily applicable to infinite volume equilibrium matter as well as to finite size pre-equilibrium systems in nuclear collisions.

Bootstrap percolation is the process on a graph where, given an initial infected set, vertices with at least r infected neighbors are infected until no new vertices can be infected. A set percolates if it infects all the vertices of the graph, and a percolating set is minimal if no proper subset percolates. We consider the question for trees. We describe an O(n) algorithm for computing the larg...

Let Hn be the hypercube {0, 1} n, and let Hn,p denote the same graph with Bernoulli bond percolation with parameter p = n−α. It is shown that at α = 1/2 there is a phase transition for the metric distortion between Hn and Hn,p. For α < 1/2, asymptotically there is a map from Hn to Hn,p with constant distortion (depending only on α). For α > 1/2 the distortion tends to infinity as a power of n. ...

We derive an Ornstein-Zernike asymptotic formula for the decay of the two point finite connectivity function τ f x,y(p) of the Bernoulli bond percolation process on Z , along the principal directions, for d ≥ 3, and for supercritical values of p sufficiently near to p = 1.

We investigate a limit theorem on traversable length inside semi-cylinder in the 2-dimensional supercritical Bernoulli bond percolation, which gives an extension of Theorem 2 in [5]. This type of limit theorems was originally studied for the extinction time for the 1-dimensional contact process on a finite interval in [10]. Actually, our main result Theorem 2.1 is stated under a rather general ...

We study a continuous-time random walk on the d-dimensional lattice, subject to a drift and an attraction to large clusters of a subcritical Bernoulli site percolation. We find two distinct regimes: a ballistic one, and a subballistic one taking place when the attraction is strong enough. We identify the speed in the former case, and the algebraic rate of escape in the latter case. Finally, we ...

The connectivity of wireless network is critical for effective and efficient communications. Yet, it is challenged by mobility, transmission range and density. In terms of increasing node density, there is a transition phase of connectivity when a network starts transforming from partition to connected state. In this paper, we study the upperand lowerbounds of critical density for the transitio...

Does there exist a Lipschitz injection of Z into the open set of a site percolation process on Z, if the percolation parameter p is sufficiently close to 1? We prove a negative answer when d = D, and also when d ≥ 2 if the Lipschitz constant M is required to be 1. Earlier work of Dirr, Dondl, Grimmett, Holroyd, and Scheutzow yields a positive answer for d < D and M = 2. As a result, the above q...