77 research outputs found
Routing on trees
We consider three different schemes for signal routing on a tree. The
vertices of the tree represent transceivers that can transmit and receive
signals, and are equipped with i.i.d. weights representing the strength of the
transceivers. The edges of the tree are also equipped with i.i.d. weights,
representing the costs for passing the edges. For each one of our schemes, we
derive sharp conditions on the distributions of the vertex weights and the edge
weights that determine when the root can transmit a signal over arbitrarily
large distances
The two-type Richardson model with unbounded initial configurations
The two-type Richardson model describes the growth of two competing
infections on and the main question is whether both infection
types can simultaneously grow to occupy infinite parts of . For
bounded initial configurations, this has been thoroughly studied. In this
paper, an unbounded initial configuration consisting of points
in the hyperplane is
considered. It is shown that, starting from a configuration where all points in
\mathcal{H} {\mathbf{0}\} are type 1 infected and the origin is
type 2 infected, there is a positive probability for the type 2 infection to
grow unboundedly if and only if it has a strictly larger intensity than the
type 1 infection. If, instead, the initial type 1 infection is restricted to
the negative -axis, it is shown that the type 2 infection at the origin
can also grow unboundedly when the infection types have the same intensity.Comment: Published in at http://dx.doi.org/10.1214/07-AAP440 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Bipartite stable Poisson graphs on R
Let red and blue points be distributed on according to two
independent Poisson processes and and let each red
(blue) point independently be equipped with a random number of half-edges
according to a probability distribution (). We consider
translation-invariant bipartite random graphs with vertex classes defined by
the point sets of and , respectively, generated by a
scheme based on the Gale-Shapley stable marriage for perfectly matching the
half-edges. Our main result is that, when all vertices have degree 2 almost
surely, then the resulting graph does not contain an infinite component. The
two-color model is hence qualitatively different from the one-color model,
where Deijfen, Holroyd and Peres have given strong evidence that there is an
infinite component. We also present simulation results for other degree
distributions
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