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