8,521 research outputs found
Scaling window for mean-field percolation of averages
For a complete graph of size , assign each edge an i.i.d. exponential
variable with mean . For , consider the length of the longest
path whose average weight is at most . It was shown by Aldous (1998)
that the length is of order for and of order
for . Aldous (2003) posed the question on detailed
behavior at and near criticality . In particular, Aldous asked
whether there exist scaling exponents such that for within
of order , the length for the longest path of average
weight at most has order .
We answer this question by showing that the critical behavior is far richer:
For around within a window of
with a small absolute constant , the longest path is of order . Furthermore, for with
a large absolute constant, the longest path is at least of length a
polynomial in . An interesting consequence of our result is the existence of
a second transition point in . In addition, we demonstrate a smooth transition from subcritical to
critical regime. Our results were not known before even in a heuristic sense.Comment: 17pages. Minor revision upon previous version. To appear in Annals of
Probabilit
Mixing under monotone censoring
We initiate the study of mixing times of Markov chain under monotone
censoring. Suppose we have some Markov Chain on a state space with
stationary distribution and a monotone set . We
consider the chain which is the same as the chain started at some except that moves of of the form where and are {\em censored} and replaced by the move . If is
ergodic and is connected, the new chain converges to conditional on
. In this paper we are interested in the mixing time of the chain in
terms of properties of and . Our results are based on new connections
with the field of property testing. A number of open problems are presented.Comment: 6 page
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