Stochastic Threshold Growth
Thomas Bohman
MIT
September 22,
4:15pm
refreshments at 3:45pm
2338
ABSTRACT

Consider the following random model for a growing droplet in
the plane. The model has 3 parameters: an update
probability $p$, a threshold
$\theta$ and a neighborhood of the origin $N$. This neighborhood is
a finite symmetric subset of the 2dimensional integer lattices, and
the neighborhood of an arbitrary $x$ in the lattice is $x+N$.
The droplet is a random sequence $ A_0 \subseteq A_1 \subseteq \dotsm $
of subsets of the two dimensional integer lattice generated as follows:
a point $x$ joins $A_{i+1}$ with probability $p$ if the intersection
of $A_i$ and the neighborhood of $x$ is at least $\theta $.
This process, known as stochastic threshold growth, generalizes
Eden's model, Richardson's rule and discrete threshold growth.
It is known that droplets generated by Eden's model, Richardson's
rule or discrete threshold growth converge to limiting shapes.
In this talk, we discuss a combinatorial lemma that is central
to the proof that limiting shapes exist for stochastic
threshold growth.

Speaker's Contact Info: bohman(atsign)math.mit.edu
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