48 research outputs found
The random interchange process on the hypercube
We prove the occurrence of a phase transition accompanied by the emergence of
cycles of diverging lengths in the random interchange process on the hypercube.Comment: 8 page
On the largest eigenvalue of a sparse random subgraph of the hypercube
We consider a sparse random subraph of the -cube where each edge appears
independently with small probability . In the most
interesting regime when is not exponentially small we prove that the
largest eigenvalue of the graph is asymtotically equal to the square root of
the maximum degree
Large components in random induced subgraphs of n-cubes
In this paper we study random induced subgraphs of the binary -cube,
. This random graph is obtained by selecting each -vertex with
independent probability . Using a novel construction of
subcomponents we study the largest component for
, where , . We prove that there exists a.s. a unique largest
component . We furthermore show that , and for , holds.
This improves the result of \cite{Bollobas:91} where constant is
considered. In particular, in case of , our
analysis implies that a.s. a unique giant component exists.Comment: 18 Page
Random induced subgraphs of Cayley graphs induced by transpositions
In this paper we study random induced subgraphs of Cayley graphs of the
symmetric group induced by an arbitrary minimal generating set of
transpositions. A random induced subgraph of this Cayley graph is obtained by
selecting permutations with independent probability, . Our main
result is that for any minimal generating set of transpositions, for
probabilities where , a random induced subgraph has a.s. a unique
largest component of size , where
is the survival probability of a specific branching process.Comment: 18 pages, 1 figur