We investigate percolation on the graph of the direct product T × Z of a regular tree T and the line Z, in which each 'tree' edge is open with probability τ and each 'line' edge with probability λ. There are three non-trivial phases, corresponding to the existence of 0, ∞, and 1 infinite open clusters. Such results may be obtained also for the graph T × Z d where d ≥ 2.

We consider the following random process on the complete graph: repeatedly draw edges (with replacement) and with probability p assign the vertices of the edge blue and with probability 1 − p assign the vertices of the edge red. This is a random walk on a state space of red/blue colorings of the complete graph and so has a stationary distribution. We derive this stationary distribution as well ...

Given an integer m, a probability measure ν on [0, 1], a process X and a real function g, we define the m-order ν-integral having as integrator X and as integrand g(X). In the case of the fractional Brownian motion B , for any locally bounded function g, the corresponding integral vanishes for all odd indices m > 1 2H and any symmetric ν. One consequence is an Itô-Stratonovich type expansion fo...

For a graph G, a strong edge coloring of G is an edge coloring in which every color class is an induced matching. The strong chromatic index of G, χs(G), is the smallest number of colors in a strong edge coloring of G. In [9], Z. Palka proved that if p = p(n) = Θ(n−1), then with high probability, χs(G(n, p)) = O(∆(G(n, p))). Recently in [12], V. Vu proved that if n−1(lnn)1+δ ≤ p = p(n) ≤ n−ε fo...

For a graph G = (V,E) of even order, a partition (V1, V2) of the vertices is said to be perfectly balanced if |V1| = |V2| and the numbers of edges in the subgraphs induced by V1 and V2 are equal. For a base graph H define a random graph G(H, p) by turning every non-edge of H into an edge and every edge of H into a nonedge independently with probability p. We show that for any constant ǫ there i...

We study a problem on percolation on product graphs G×K2. Here G is any finite graph and K2 consists of two vertices {0, 1} connected by an edge. In edge percolation every edge in G × K2 is present with probability p. In [3] Olle Häggström stated a conjecture (which he claimed to be folklore) that for all G and p the probability that (u, 0) is in the same component as (v, 0) is greater than the...

The pre-peak in the x-ray-absorption near-edge structure ~XANES! of CuFeS2 is studied by exploring the associated photoelectron probability density as well as by investigating the cluster size effect. All significant features of the Cu, Fe, and S K-edge experimental spectra and of the S L2,3-edge spectrum are successfully reproduced by the real-space multiple-scattering ~RSMS! calculations with...

We introduce a collection of complex networks generated by a combination of preferential attachment and a previously unexamined process of “splitting” nodes of degree k into k nodes of degree 1. Four networks are considered, each evolves at each time step by either preferential attachment, with probability p, or splitting with probability 1 − p. Two methods of attachment are considered; first, ...

Let G be a given graph (modelling a communication network) which we assume suuers from static edge faults: That is we let each edge of G be present independently with probability p (or absent with fault probability f = 1 ? p). In particular we are interested in ro-bustness results for the case that the graph G itself is a random member of the class of all regular graphs with given degree. Our r...

We analyze a model of fault-tolerant systems in a probabilistic setting. The model has been introduced under the name of “sabotage games”. A reachability problem over graphs is considered, where a “Runner” starts from a vertex u and seeks to reach some vertex in a target set F while, after each move, the adversary “Blocker” deletes one edge. Extending work by Löding and Rohde (who showed PSpace...