Janko Gravner
Home page:
http://www.math.ucdavis.edu/~gravner/
Position: Professor
Year joining UC Davis: 1992
Degree: Ph.D., 1991, University of Wisconsin, Madison
Refereed publications: Via
Math Reviews
Recent publications: Via
math arXiv
Professor Janko Gravner studies cellular automata
theory with emphasis on probabilistic problems. He
has analyzed various cellular automata with random initial conditions or random
choices in the transition rules. It is known that many cellular automata
systems, most notably Conway's Game of Life, are intractibly complicated
in the sense that they can simulate a Turing machine or a digital
computer. At the same time, many cellular systems model physical processes
such as phase transitions in materials (boiling and crystallization) and
biological processes such as the spread of disease or organization of living
cells. Many of these models are not of the intractibly complicated type and
seems to have predictable aggregate behavior. Professor Gravner strives to
combine computer experiments with rigorous investigations to analyze such
dynamics.
One of the simplest cellular automata is threshold growth, in which at
each step in time, a finite subset of cells in a lattice gains a new lattice
point as a member if sufficiently many neighbors in a standard neighborhood
have already been accepted. Gravner and Griffeath [2] [3] proved that, under
natural conditions, a finite set evolves under threshold growth to a
characteristic shape and developed basic nucleation and interaction
theory.
Threshold growth, in turn, can be used to model more complicated cellular
automata [5]. A closely related family of automata are threshold vote
automata, in which each cell may be undecided (quiescent) or may be moved to
adopt any one of a finite list of opinions if enough neighboring cells hold the
same opinion. Gravner and Griffeath showed that, in a variety of cases,
threshold vote automata develop domains of single opinion which then meet and
form stand-offs along boundaries. They also found a number of other such
conditions where this behavior clearly holds experimentally.
 Another related collection of automata are collectively called the
Greenberg-Hastings model [1], which involves excited and rested states,
and an ordered list of recovering states in between. Excited cells infect
neighboring rested cells, and otherwise each cell recovers one step at a time.
Frisch, Gravner, and Griffeath showed that states in this model typically
evolve into progressing spirals [1] or expanding rings [4].
Selected publications
[1] Metastability in the Greenberg-Hastings model (with R. Fisch and D.
Griffeath), Ann. Appl. Probab. 3 (1993), 935-967.
[2] The boundary of iterates in Euclidean growth models, Trans. Amer. Math.
Soc. 348 (1996), 4549-4559.
[3] First passage times for the threshold growth dynamics on Z^2 (with D.
Griffeath), Ann. Probab. 24 (1996), 1752-1778.
[4] Percolation times in two-dimensional models for excitable media,
Electron. J. Probab. 1 (1996), #12.
[5] Multitype threshold voter model and convergence to Poisson-Voronoi
tessellation (with D. Griffeath), Ann. Appl. Probab. 7 (1997), 615-647.
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