A percolation theory based on variation of conductive filler fraction has been widely used to explain the behavior of conductive composite materials under both small and large deformation conditions. However, it typically fails in properly analyzing the materials under the large deformation since the assumption may not be valid in such a case. Therefore, we proposed a new three-dimensional perc...

In this paper, we derive theoretical bounds for the long-term influence of a node in an Independent Cascade Model (ICM). We relate these bounds to the spectral radius of a particular matrix and show that the behavior is sub-critical when this spectral radius is lower than 1. More specifically, we point out that, in general networks, the sub-critical regime behaves in O( √ n) where n is the size...

The areas under workload process and under queuing process in a single server queue over the busy period have many applications not only in queuing theory but also in risk theory or percolation theory. We focus here on the tail behaviour of distribution of these two integrals. We present various open problems and conjectures, which are supported by partial results for some special cases.

I’d like to take a moment to follow up on our observations about percolation in real-world graphs from last lecture. First, I should say that the phenomenon that we observed–that these graphs did not evidence a sharp threshold–seems to be pretty well understood. Newman (the book) has an analysis of situtations like this under the configuration model, which I’ll explain in a moment, but under no...

One of the most well known examples of phenomena that introduces and motivates the study of continuum percolation is the process of the ground getting wet during a period of rain. At each point hit by a raindrop, one sees a circular wet patch. Right after the rain begins to fall what one sees is a small wet region inside a large dry region. At some instant, so many raindrops have hit the ground...

The vacant set of random interlacements at level u > 0, introduced in [8], is a percolation model onZ, d ≥ 3which arises as the set of sites avoided by a Poissonian cloud of doubly infinite trajectories, where u is a parameter controlling the density of the cloud. It was proved in [6, 8] that for any d ≥ 3 there exists a positive and finite threshold u∗ such that if u < u∗ then the vacant set p...

A useful result about leftmost and rightmost paths in two dimensional bond percolation is proved. This result was introduced without proof in [5] in the context of the contact process in continuous time. As discussed here, it also holds for several related models, including the discrete time contact process and two dimensional site percolation. Among the consequences are a natural monotonicity ...

Let H denote a collection of subsets of {1, 2, . . . , n}, and assign independent random variables uniformly distributed over [0, 1] to the n elements. Declare an element p-present if its corresponding value is at most p. In this paper, we quantify how much the observation of the r-present (r > p) set of elements affects the probability that the set of p-present elements is contained in H. In t...

It is well-known in percolation theory (and intuitively plausible) that two events of the form “there is an open path from s to a” are positively correlated. We prove the (not intuitively obvious) fact that this is still true if we condition on an event of the form “there is no open path from s to t”. 1991 Mathematics Subject Classification: 05C99 60C05 60K35.

In 1961 Gilbert defined a model of continuum percolation in which points are placed in the plane according to a Poisson process of density one, and two are joined if one lies within a disc of area A about the other. We prove some good bounds on the critical area Ac for percolation in this model. The proof is in two parts: first we give a rigorous reduction of the problem to a finite problem, an...