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abstract:
Let
G=(V,E) be an infinite connected graph in which a distinguished vertex,
the origin, is chosen. The edges of G are assigned independent uniform
random variables on [0,1], called weights. The invasion percolation
cluster of the origin on G is defined as the limit of an increasing
sequence (G_n) of connected sub-graphs of G as follows. Define G_0 to
be the origin. Given G_n = (V_n,E_n), edge set E_{n+1} is obtained from
E_n by adding to it the edge from the boundary of G_n with the smallest
weight. Let G_{n+1} be the graph induced by the edge set E_{n+1}.
The first outlet is the invaded edge of largest weight. This edge
exists for the invasion on the graphs that we consider in this talk.
For k>1, the kth outlet is defined as the edge invaded after the
(k-1)st outlet of largest weight. The graph induced by the edges
invaded between the (k-1)st and the kth outlets is called the kth pond.
The ponds are finite connected sub-graphs of G, and the invasion
percolation cluster can be viewed as a chain of ponds separated by the
outlets.
In this talk, we consider invasion percolation on two-dimensional
lattices.
We give some basic relations between invasion percolation and critical
Bernoulli percolation. We use these relations to prove almost sure
upper and lower bounds on the number of outlets within distance n from
the origin. We use these bounds to show that, almost surely, (a) the
ponds grow exponentially fast and (b) the weights of the outlets decay
exponentially fast to the percolation threshold.
(Joint work with Michael Damron.)
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abstract:
Consider
the following growth model on Z^2. Initially, only the origin is
occupied and all other sites are vacant. The cluster of occupied sites
is repeatedly expanded by starting a random walk in the origin and
occupying the first vacant site it visits. This model was originally
studied in the case where the random walk is simple random walk on the
integer lattice. Here we base the model on a wide class of random walks
enjoying a uniform layering property on the lattice, which simplifies
many of the proofs.
In the first part of the seminar, we will see that for any uniformly
layered walk, the limit shape of the occupied cluster is a diamond. In
the second part we will specialize to uniformly layered walks which
have a directional bias towards or away from the origin. We will study
how the order of the fluctuations of the occupied cluster around its
limit shape depends on the directional bias, and conjecture that an
abrupt change takes place as the directional bias tends from outward to
inward.
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