5 research outputs found
Percolation of worms
We introduce a new correlated percolation model on the -dimensional
lattice called the random length worms model. Assume given a
probability distribution on the set of positive integers (the length
distribution) and (the intensity parameter). From each site
of we start independent simple random walks
with this length distribution. We investigate the connectivity properties of
the set of sites visited by this cloud of random walks. It is
easy to show that if the second moment of the length distribution is finite
then undergoes a percolation phase transition as varies.
Our main contribution is a sufficient condition on the length distribution
which guarantees that percolates for all if .
E.g., if the probability mass function of the length distribution is
for some and then percolates for
all . Note that the second moment of this length distribution is only
"barely" infinite. In order to put our result in the context of earlier results
about similar models (e.g., finitary random interlacements, loop percolation,
Poisson Boolean model, ellipses percolation, etc.), we define a natural family
of percolation models called the Poisson zoo and argue that the percolative
behaviour of the random length worms model is quite close to being "extremal"
in this family of models.Comment: 50 page
Random cherry graphs
Due to the popularity of randomly evolving graph processes, there exists a randomized version of many recursively defined graph models. This is also the case with the cherry tree, which was introduced by Bukszar and Prekopa to improve Bonferroni type upper bounds on the probability of the union of random events. Here we consider a substantially extended random analogue of that model, embedding it into a general time-dependent branching process
Moments of general time dependent branching processes with applications
In this paper, we give sufficient conditions for a Crump-Mode-Jagers process
to be bounded in for a given . This result is then applied to a
recent random graph process motivated by pairwise collaborations and driven by
time-dependent branching dynamics. We show that the maximal degree has the same
rate of increase as the degree process of a fixed vertex.Comment: 12 page