DIMACS Discrete Math/Theory of Computing Seminar
A Discrete Model for Crystal Growth in the Plane
- Tom Bohman
- Rutgers University
- CoRE Building Room 431
- Busch Campus, Rutgers University
- 4:30 PM
- Tuesday, March 5, 1996
In the discrete threshold model for crystal growth in the plane
we begin with some subset A_0 of Z^2 of seed crystals and observe
crystal growth over time by generating a sequence of subsets A_0
\subset A_1 \subset A_2 \subset ... of Z^2 by a deterministic rule.
This rule is as follows: a site crystallizes when a threshold number
of crystallized points appear in the site's prescribed neighborhood.
The dynamics (the choice of neighborhood and threshold) are said to be
omnivorous if for any finite set of seed crystals for which the
crystal never stops growing the crystal eventually occupies all of Z^2.
In this talk we prove that the dynamics are omivorous when the
neighborhood is a box. This result has important implications in the
study of the first passage time when A_0 is chosen randomly with
a sparse Bernoulli density and in the study of the limiting shape to
which A_n/n converges.
Document last modified on February 28, 1996