11,932 research outputs found
Graph diameter in long-range percolation
We study the asymptotic growth of the diameter of a graph obtained by adding
sparse "long" edges to a square box in . We focus on the cases when an
edge between and is added with probability decaying with the Euclidean
distance as when . For we show
that the graph diameter for the graph reduced to a box of side scales like
where . In particular, the
diameter grows about as fast as the typical graph distance between two vertices
at distance . We also show that a ball of radius in the intrinsic metric
on the (infinite) graph will roughly coincide with a ball of radius
in the Euclidean metric.Comment: 17 pages, extends the results of arXiv:math.PR/0304418 to graph
diameter, substantially revised and corrected, added a result on volume
growth asymptoti
Long-range first-passage percolation on the complete graph
We study a geometric version of first-passage percolation on the complete
graph, known as long-range first-passage percolation. Here, the vertices of the
complete graph are embedded in the -dimensional torus
, and each edge is assigned an independent transmission time
, where is a rate-one exponential
random variable associated with the edge ,
denotes the torus-norm, and is a parameter. We are interested in
the case , which corresponds to the instantaneous percolation
regime for long-range first-passage percolation on studied by
Chatterjee and Dey, and which extends first-passage percolation on the complete
graph (the case) studied by Janson. We consider the typical
distance, flooding time, and diameter of the model. Our results show a
-type result, akin to first-passage percolation on the complete graph as
shown by Janson. The results also provide a quantitative perspective to the
qualitative results observed by Chatterjee and Dey on .Comment: 16 page
Transience and recurrence of random walks on percolation clusters in an ultrametric space
We study existence of percolation in the hierarchical group of order ,
which is an ultrametric space, and transience and recurrence of random walks on
the percolation clusters. The connection probability on the hierarchical group
for two points separated by distance is of the form , with , non-negative constants , and . Percolation was proved in Dawson and Gorostiza
(2013) for , with
. In this paper we improve the result for the critical case by
showing percolation for . We use a renormalization method of the type
in the previous paper in a new way which is more intrinsic to the model. The
proof involves ultrametric random graphs (described in the Introduction). The
results for simple (nearest neighbour) random walks on the percolation clusters
are: in the case the walk is transient, and in the critical case
, there exists a critical
such that the walk is recurrent for and transient for
. The proofs involve graph diameters, path lengths, and
electric circuit theory. Some comparisons are made with behaviours of random
walks on long-range percolation clusters in the one-dimensional Euclidean
lattice.Comment: 27 page
On the scaling of the chemical distance in long-range percolation models
We consider the (unoriented) long-range percolation on Z^d in dimensions
d\ge1, where distinct sites x,y\in Z^d get connected with probability
p_{xy}\in[0,1]. Assuming p_{xy}=|x-y|^{-s+o(1)} as |x-y|\to\infty, where s>0
and |\cdot| is a norm distance on Z^d, and supposing that the resulting random
graph contains an infinite connected component C_{\infty}, we let D(x,y) be the
graph distance between x and y measured on C_{\infty}. Our main result is that,
for s\in(d,2d), D(x,y)=(\log|x-y|)^{\Delta+o(1)},\qquad x,y\in C_{\infty},
|x-y|\to\infty, where \Delta^{-1} is the binary logarithm of 2d/s and o(1) is a
quantity tending to zero in probability as |x-y|\to\infty. Besides its interest
for general percolation theory, this result sheds some light on a question that
has recently surfaced in the context of ``small-world'' phenomena. As part of
the proof we also establish tight bounds on the probability that the largest
connected component in a finite box contains a positive fraction of all sites
in the box.Comment: Published at http://dx.doi.org/10.1214/009117904000000577 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Structures in supercritical scale-free percolation
Scale-free percolation is a percolation model on which can be
used to model real-world networks. We prove bounds for the graph distance in
the regime where vertices have infinite degrees. We fully characterize
transience vs. recurrence for dimension 1 and 2 and give sufficient conditions
for transience in dimension 3 and higher. Finally, we show the existence of a
hierarchical structure for parameters where vertices have degrees with infinite
variance and obtain bounds on the cluster density.Comment: Revised Definition 2.5 and an argument in Section 6, results are
unchanged. Correction of minor typos. 29 pages, 7 figure
Long-range percolation on the hierarchical lattice
We study long-range percolation on the hierarchical lattice of order ,
where any edge of length is present with probability
, independently of all other edges. For fixed
, we show that the critical value is non-trivial if
and only if . Furthermore, we show uniqueness of the infinite
component and continuity of the percolation probability and of
as a function of . This means that the phase diagram
of this model is well understood.Comment: 24 page
- …