26 research outputs found
Bootstrap percolation on the Hamming torus with threshold 2
This paper analyzes various questions pertaining to bootstrap percolation on
the -dimensional Hamming torus where each node is open with probability
and the percolation threshold is 2. For each we find the critical
exponent for the event that a -dimensional subtorus becomes open and
compute the limiting value of its probability under the critical scaling. For
even , we use the Chen-Stein method to show that the number of
-dimensional subtori that become open can be approximated by a Poisson
random variable.Comment: Various revision
Expected size of a tree in the fixed point forest
We study the local limit of the fixed-point forest, a tree structure
associated to a simple sorting algorithm on permutations. This local limit can
be viewed as an infinite random tree that can be constructed from a Poisson
point process configuration on . We generalize this random
tree, and compute the expected size and expected number of leaves of a random
rooted subtree in the generalized version. We also obtain bounds on the
variance of the size.Comment: 14 page
Asymptotic distribution of fixed points of pattern-avoiding involutions
For a variety of pattern-avoiding classes, we describe the limiting
distribution for the number of fixed points for involutions chosen uniformly at
random from that class. In particular we consider monotone patterns of
arbitrary length as well as all patterns of length 3. For monotone patterns we
utilize the connection with standard Young tableaux with at most rows and
involutions avoiding a monotone pattern of length . For every pattern of
length 3 we give the bivariate generating function with respect to fixed points
for the involutions that avoid that pattern, and where applicable apply tools
from analytic combinatorics to extract information about the limiting
distribution from the generating function. Many well-known distributions
appear.Comment: 16 page
The threshold for jigsaw percolation on random graphs
Jigsaw percolation is a model for the process of solving puzzles within a
social network, which was recently proposed by Brummitt, Chatterjee, Dey and
Sivakoff. In the model there are two graphs on a single vertex set (the
`people' graph and the `puzzle' graph), and vertices merge to form components
if they are joined by an edge of each graph. These components then merge to
form larger components if again there is an edge of each graph joining them,
and so on. Percolation is said to occur if the process terminates with a single
component containing every vertex. In this note we determine the threshold for
percolation up to a constant factor, in the case where both graphs are
Erd\H{o}s--R\'enyi random graphs.Comment: 13 page
Scaling limits of permutations avoiding long decreasing sequences
We determine the scaling limit for permutations conditioned to have longest
decreasing subsequence of length at most . These permutations are also said
to avoid the pattern and they can be written as a union of
increasing subsequences. We show that these increasing subsequences can be
chosen so that, after proper scaling, and centering, they converge in
distribution. As the size of the permutations tends to infinity, the
distribution of functions generated by the permutations converges to the
eigenvalue process of a traceless Hermitian Brownian bridge.Comment: 58 pages, 10 figures, minor edits and additional figures adde