176 research outputs found
Exponential growth of ponds in invasion percolation on regular trees
In invasion percolation, the edges of successively maximal weight (the
outlets) divide the invasion cluster into a chain of ponds separated by
outlets. On the regular tree, the ponds are shown to grow exponentially, with
law of large numbers, central limit theorem and large deviation results. The
tail asymptotics for a fixed pond are also studied and are shown to be related
to the asymptotics of a critical percolation cluster, with a logarithmic
correction
Scaling limit of the invasion percolation cluster on a regular tree
We prove existence of the scaling limit of the invasion percolation cluster
(IPC) on a regular tree. The limit is a random real tree with a single end. The
contour and height functions of the limit are described as certain diffusive
stochastic processes. This convergence allows us to recover and make precise
certain asymptotic results for the IPC. In particular, we relate the limit of
the rescaled level sets of the IPC to the local time of the scaled height
function.Comment: Published in at http://dx.doi.org/10.1214/11-AOP731 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Invasion percolation on regular trees
We consider invasion percolation on a rooted regular tree. For the infinite
cluster invaded from the root, we identify the scaling behavior of its
-point function for any and of its volume both at a given height
and below a given height. We find that while the power laws of the scaling are
the same as for the incipient infinite cluster for ordinary percolation, the
scaling functions differ. Thus, somewhat surprisingly, the two clusters behave
differently; in fact, we prove that their laws are mutually singular. In
addition, we derive scaling estimates for simple random walk on the cluster
starting from the root. We show that the invasion percolation cluster is
stochastically dominated by the incipient infinite cluster. Far above the root,
the two clusters have the same law locally, but not globally. A key ingredient
in the proofs is an analysis of the forward maximal weights along the backbone
of the invasion percolation cluster. These weights decay toward the critical
value for ordinary percolation, but only slowly, and this slow decay causes the
scaling behavior to differ from that of the incipient infinite cluster.Comment: Published in at http://dx.doi.org/10.1214/07-AOP346 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Degree distribution of shortest path trees and bias of network sampling algorithms
In this article, we explicitly derive the limiting degree distribution of the
shortest path tree from a single source on various random network models with
edge weights. We determine the asymptotics of the degree distribution for large
degrees of this tree and compare it to the degree distribution of the original
graph. We perform this analysis for the complete graph with edge weights that
are powers of exponential random variables (weak disorder in the stochastic
mean-field model of distance), as well as on the configuration model with
edge-weights drawn according to any continuous distribution. In the latter, the
focus is on settings where the degrees obey a power law, and we show that the
shortest path tree again obeys a power law with the same degree power-law
exponent. We also consider random -regular graphs for large , and show
that the degree distribution of the shortest path tree is closely related to
the shortest path tree for the stochastic mean-field model of distance. We use
our results to shed light on an empirically observed bias in network sampling
methods. This is part of a general program initiated in previous works by
Bhamidi, van der Hofstad and Hooghiemstra [Ann. Appl. Probab. 20 (2010)
1907-1965], [Combin. Probab. Comput. 20 (2011) 683-707], [Adv. in Appl. Probab.
42 (2010) 706-738] of analyzing the effect of attaching random edge lengths on
the geometry of random network models.Comment: Published at http://dx.doi.org/10.1214/14-AAP1036 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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