We prove the existence of a large complete subgraph w.h.p. in a preferential attachment random graph process with an edge-step. That is, we consider a dynamic stochastic process for constructing a graph in which at each step we independently decide, with probability p ∈ (0, 1), whether the graph receives a new vertex or a new edge between existing vertices. The connections are then made according to a preferential attachment rule. We prove that the random graph G t produced by this so-called GLP (Generalized linear preferential) model at time t contains a complete subgraph whose vertex set cardinality is given by t α , where α = (1 − ε) 1−p 2−p , for any small ε > 0 asymptotically almost surely.

It has been recently understood [8,23,27] that for a general class of percolation models on Z d satisfying suitable decoupling inequalities, which includes i.a. Bernoulli percolation, random interlacements and level sets of the Gaussian free field, large scale geometry of the unique infinite cluster in strongly percolative regime is qualitatively the same; in particular, the random walk on the infinite cluster satisfies the quenched invariance principle, Gaussian heat-kernel bounds and local CLT.In this paper we consider the random walk loop soup on Z d in dimensions d ≥ 3. An interesting aspect of this model is that despite its similarity and connections to random interlacements and the Gaussian free field, it does not fall into the above mentioned general class of percolation models, since it does not satisfy the required decoupling inequalities.We identify weaker (and more natural) decoupling inequalities and prove that (a) they do hold for the random walk loop soup and (b) all the results about the large scale geometry of the infinite percolation cluster proved for the above mentioned class of models hold also for models that satisfy the weaker decoupling inequalities. Particularly, all these results are new for the vacant set of the random walk loop soup. (The range of the random walk loop soup has been addressed by Chang [6] by a model specific approximation method, which does not apply to the vacant set.)Finally, we prove that the strongly supercritical regime for the vacant set of the random walk loop soup is non-trivial. It is expected, but open at the moment, that the strongly supercritical regime coincides with the whole supercritical regime.

In this work we investigate a preferential attachment model whose parameter is a function f : N → [0, 1] that drives the asymptotic proportion between the numbers of vertices and edges of the graph. We investigate topological features of the graphs, proving general bounds for the diameter and the clique number. Our results regarding the diameter are sharp when f is a regularly varying function at infinity with strictly negative index of regular variation −γ. For this particular class, we prove a characterization for the diameter that depends only on −γ. More specifically, we prove that the diameter of such graphs is of order 1/γ with high probability, although its vertex set order goes to infinity polynomially. Sharp results for the diameter for a wide class of slowly varying functions are also obtained. The almost sure convergence for the properly normalized logarithm of the clique number of the graphs generated by slowly varying functions is also proved.

Abstract. We prove a conditional decoupling inequality for the model of random interlacements in dimension d ≥ 3: the conditional law of random interlacements on a box (or a ball) A 1 given the (not very "bad") configuration on a "distant" set A 2 does not differ a lot from the unconditional law. The main method we use is a suitable modification of the soft local time method of [13], that allows dealing with conditional probabilities.

We consider oriented long-range percolation on a graph with vertex set Z d × Z + and directed edges of the form (x, t), (x + y, t + 1) , for x, y in Z d and t ∈ Z + . Any edge of this form is open with probability p y , independently for all edges. Under the assumption that the values p y do not vanish at infinity, we show that there is percolation even if all edges of length more than k are deleted, for k large enough. We also state the analogous result for a long-range contact process on Z d .