18 research outputs found
A percolation process on the binary tree where large finite clusters are frozen
We study a percolation process on the planted binary tree, where clusters
freeze as soon as they become larger than some fixed parameter N. We show that
as N goes to infinity, the process converges in some sense to the frozen
percolation process introduced by Aldous. In particular, our results show that
the asymptotic behaviour differs substantially from that on the square lattice,
on which a similar process has been studied recently by van den Berg, de Lima
and Nolin.Comment: 11 page
On the size of the largest cluster in 2D critical percolation
We consider (near-)critical percolation on the square lattice. Let M_n be the
size of the largest open cluster contained in the box [-n,n]^2, and let pi(n)
be the probability that there is an open path from O to the boundary of the
box. It is well-known that for all 0< a < b the probability that M_n is smaller
than an^2 pi(n) and the probability that M_n is larger than bn^2 pi(n) are
bounded away from 0 as n tends to infinity. It is a natural question, which
arises for instance in the study of so-called frozen-percolation processes, if
a similar result holds for the probability that M_n is between an^2 pi(n) and
bn^2 pi(n). By a suitable partition of the box, and a careful construction
involving the building blocks, we show that the answer to this question is
affirmative. The `sublinearity' of 1/pi(n) appears to be essential for the
argument.Comment: 12 pages, 3 figures, minor change
Two-dimensional volume-frozen percolation: exceptional scales
We study a percolation model on the square lattice, where clusters "freeze"
(stop growing) as soon as their volume (i.e. the number of sites they contain)
gets larger than N, the parameter of the model. A model where clusters freeze
when they reach diameter at least N was studied in earlier papers. Using volume
as a way to measure the size of a cluster - instead of diameter - leads, for
large N, to a quite different behavior (contrary to what happens on the binary
tree, where the volume model and the diameter model are "asymptotically the
same"). In particular, we show the existence of a sequence of "exceptional"
length scales.Comment: 20 pages, 2 figure
Boundary rules and breaking of self-organized criticality in 2D frozen percolation
We study frozen percolation on the (planar) triangular lattice, where connected components stop growing (“freeze”) as soon as their “size” becomes at least N, for some parameter N ≥ 1. The size of a connected component can be measured in several natural ways, and we consider the two particular cases of diameter and volume (i.e. number of sites).
Diameter-frozen and volume-frozen percolation have been studied in previous works ([5, 15] and [6, 4], resp.), and they display radically different behaviors. These works adopt the rule that the boundary of a frozen cluster stays vacant forever, and we investigate the influence of these “boundary rules” in the present paper. We prove the (somewhat surprising) result that they strongly matter in the diameter case, and we discuss briefly the volume case
Boundary rules and breaking of self-organized criticality in 2D frozen percolation
We study frozen percolation on the (planar) triangular lattice, where connected components stop growing ("freeze") as soon as their "size" becomes at least N, for some parameter N ≥ 1. The size of a connected component can be measured in several natural ways, and we consider the two particular cases of diameter and volume (i.e. number of sites).
Diameter-frozen and volume-frozen percolation have been studied in previous works ([25, 11] and [27, 26], resp.), and they display radically different behaviors. These works adopt the rule that the boundary of a frozen cluster stays vacant forever, and we investigate the influence of these "boundary conditions" in the present paper. We prove the (somewhat surprising) result that they strongly matter in the diameter case, and we discuss briefly the volume case
Percolation with constant freezing
We introduce and study a model of percolation with constant freezing (PCF)
where edges open at constant rate 1, and clusters freeze at rate \alpha
independently of their size. Our main result is that the infinite volume
process can be constructed on any amenable vertex transitive graph. This is in
sharp contrast to models of percolation with freezing previously introduced,
where the limit is known not to exist. Our interest is in the study of the
percolative properties of the final configuration as a function of \alpha. We
also obtain more precise results in the case of trees. Surprisingly the
algebraic exponent for the cluster size depends on the degree, suggesting that
there is no lower critical dimension for the model. Moreover, even for
\alpha<\alpha_c, it is shown that finite clusters have algebraic tail decay,
which is a signature of self organised criticality. Partial results are
obtained on Z^d, and many open questions are discussed.Comment: 30 pages, 8 figure