46 research outputs found
Critical percolation on random regular graphs
We show that for all the size of the largest
component of a random -regular graph on vertices around the percolation
threshold is , with high probability. This extends
known results for fixed and for , confirming a prediction of
Nachmias and Peres on a question of Benjamini. As a corollary, for the largest
component of the percolated random -regular graph, we also determine the
diameter and the mixing time of the lazy random walk. In contrast to previous
approaches, our proof is based on a simple application of the switching method.Comment: 10 page
Scaling limits for critical inhomogeneous random graphs with finite third moments
We identify the scaling limits for the sizes of the largest components at
criticality for inhomogeneous random graphs when the degree exponent
satisfies . We see that the sizes of the (rescaled) components converge
to the excursion lengths of an inhomogeneous Brownian motion, extending results
of \cite{Aldo97}. We rely heavily on martingale convergence techniques, and
concentration properties of (super)martingales. This paper is part of a
programme to study the critical behavior in inhomogeneous random graphs of
so-called rank-1 initiated in \cite{Hofs09a}.Comment: Final versio
The Incipient Giant Component in Bond Percolation on General Finite Weighted Graphs
On a large finite connected graph let edges become "open" at independent
random Exponential times of arbitrary rates . Under minimal assumptions,
the time at which a giant component starts to emerge is weakly concentrated
around its mean
Sharp threshold for percolation on expanders
We study the appearance of the giant component in random subgraphs of a given
large finite graph G=(V,E) in which each edge is present independently with
probability p. We show that if G is an expander with vertices of bounded
degree, then for any c in ]0,1[, the property that the random subgraph contains
a giant component of size c|V| has a sharp threshold.Comment: Published in at http://dx.doi.org/10.1214/10-AOP610 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Random graph asymptotics on high-dimensional tori. II. Volume, diameter and mixing time
For critical bond-percolation on high-dimensional torus, this paper proves
sharp lower bounds on the size of the largest cluster, removing a logarithmic
correction in the lower bound in Heydenreich and van der Hofstad (2007). This
improvement finally settles a conjecture by Aizenman (1997) about the role of
boundary conditions in critical high-dimensional percolation, and it is a key
step in deriving further properties of critical percolation on the torus.
Indeed, a criterion of Nachmias and Peres (2008) implies appropriate bounds on
diameter and mixing time of the largest clusters. We further prove that the
volume bounds apply also to any finite number of the largest clusters. The main
conclusion of the paper is that the behavior of critical percolation on the
high-dimensional torus is the same as for critical Erdos-Renyi random graphs.
In this updated version we incorporate an erratum to be published in a
forthcoming issue of Probab. Theory Relat. Fields. This results in a
modification of Theorem 1.2 as well as Proposition 3.1.Comment: 16 pages. v4 incorporates an erratum to be published in a forthcoming
issue of Probab. Theory Relat. Field
The evolution of the cover time
The cover time of a graph is a celebrated example of a parameter that is easy
to approximate using a randomized algorithm, but for which no constant factor
deterministic polynomial time approximation is known. A breakthrough due to
Kahn, Kim, Lovasz and Vu yielded a (log log n)^2 polynomial time approximation.
We refine this upper bound, and show that the resulting bound is sharp and
explicitly computable in random graphs. Cooper and Frieze showed that the cover
time of the largest component of the Erdos-Renyi random graph G(n,c/n) in the
supercritical regime with c>1 fixed, is asymptotic to f(c) n \log^2 n, where
f(c) tends to 1 as c tends to 1. However, our new bound implies that the cover
time for the critical Erdos-Renyi random graph G(n,1/n) has order n, and shows
how the cover time evolves from the critical window to the supercritical phase.
Our general estimate also yields the order of the cover time for a variety of
other concrete graphs, including critical percolation clusters on the Hamming
hypercube {0,1}^n, on high-girth expanders, and on tori Z_n^d for fixed large
d. For the graphs we consider, our results show that the blanket time,
introduced by Winkler and Zuckerman, is within a constant factor of the cover
time. Finally, we prove that for any connected graph, adding an edge can
increase the cover time by at most a factor of 4.Comment: 14 pages, to appear in CP