17 research outputs found
Asymptotic normality of the size of the giant component in a random hypergraph
Recently, we adapted random walk arguments based on work of Nachmias and
Peres, Martin-L\"of, Karp and Aldous to give a simple proof of the asymptotic
normality of the size of the giant component in the random graph above
the phase transition. Here we show that the same method applies to the
analogous model of random -uniform hypergraphs, establishing asymptotic
normality throughout the (sparse) supercritical regime. Previously, asymptotic
normality was known only towards the two ends of this regime.Comment: 11 page
On the non-Gaussian fluctuations of the giant cluster for percolation on random recursive trees
We consider a Bernoulli bond percolation on a random recursive tree of size
, with supercritical parameter for some fixed. It
is known that with high probability, there exists then a unique giant cluster
of size G_n\sim \e^{-c}, and it follows from a recent result of Schweinsberg
\cite{Sch} that has non-gaussian fluctuations. We provide an explanation
of this by analyzing the effect of percolation on different phases of the
growth of recursive trees. This alternative approach may be useful for studying
percolation on other classes of trees, such as for instance regular trees
Exploring hypergraphs with martingales
Recently, we adapted exploration and martingale arguments of Nachmias and
Peres, in turn based on ideas of Martin-L\"of, Karp and Aldous, to prove
asymptotic normality of the number of vertices in the largest component
of the random -uniform hypergraph throughout the supercritical regime.
In this paper we take these arguments further to prove two new results: strong
tail bounds on the distribution of , and joint asymptotic normality of
and the number of edges of . These results are used in a
separate paper "Counting connected hypergraphs via the probabilistic method" to
enumerate sparsely connected hypergraphs asymptotically.Comment: 32 pages; significantly expanded presentation. To appear in Random
Structures and Algorithm