6 research outputs found
Cutting edges at random in large recursive trees
We comment on old and new results related to the destruction of a random
recursive tree (RRT), in which its edges are cut one after the other in a
uniform random order. In particular, we study the number of steps needed to
isolate or disconnect certain distinguished vertices when the size of the tree
tends to infinity. New probabilistic explanations are given in terms of the
so-called cut-tree and the tree of component sizes, which both encode different
aspects of the destruction process. Finally, we establish the connection to
Bernoulli bond percolation on large RRT's and present recent results on the
cluster sizes in the supercritical regime.Comment: 29 pages, 3 figure
On large deviation properties of Erdos-Renyi random graphs
We show that large deviation properties of Erd\"os-R\'enyi random graphs can
be derived from the free energy of the -state Potts model of statistical
mechanics. More precisely the Legendre transform of the Potts free energy with
respect to is related to the component generating function of the graph
ensemble. This generalizes the well-known mapping between typical properties of
random graphs and the limit of the Potts free energy. For
exponentially rare graphs we explicitly calculate the number of components, the
size of the giant component, the degree distributions inside and outside the
giant component, and the distribution of small component sizes. We also perform
numerical simulations which are in very good agreement with our analytical
work. Finally we demonstrate how the same results can be derived by studying
the evolution of random graphs under the insertion of new vertices and edges,
without recourse to the thermodynamics of the Potts model.Comment: 38 pages, 9 figures, Latex2e, corrected and extended version
including numerical simulation result
The Giant Component Is Normal
We prove a central limit theorem for the fluctuations of the size of the Giant Component in a random graph with small edge probability