About the Particle Postcards

Main Kitchen postcard: Bootstrap percolation
 Start with a random density p of occupied sites (1's on
a background of 0's). At each update an empty site becomes occupied
if it sees at least two previously occupied sites among its four nearest
neighbors (N,S,E,W). Under these dynamics an isolated
rectangle of occupied sites cannot grow, but any occupied region that is
not convex must continue to grow. By superimposing the resulting dynamics
for varying values of p, and color coding stable regions for small
p with oranges and yellows and larger nucleating regions for large
p with blues and greens, we obtain a striking image of nucleation.
For more on the mathematics of this rule, see Aizenman, M. and Lebowitz, J.
(1988) "Metastability effects in bootstrap percolation." J. Phys. A: Math.
Gen. 21, 38013813.
Come On In postcard: A Cyclic Smoothing Process
 Choose a cyclic palette in which neighboring colors differ
imperceptibly. Our wheel consists of a smooth spectrum of greens and
blues suggestive of primordial soup. The asynchronous interaction rule is
to replace the colors of two neighboring sites by their "average," following
the shorter route around the color wheel. Starting from a random
configuration we see a simultaneous process of clustering and
selforganization. The averaging causes nearby colors to
cluster. But since there is no bias toward any region of the spectrum, the
result is a homogeneous continuous flow with a "water color" feel, marked
by vortices known as 'defects.' This simple rule is related to the
Heisenberg model of statistical physics.
Recent Additions postcard: A Directed Random Interface from Consensus
 This postcard depicts the same experiment as the previous one, except that time
progresses downward and the initial configuration at the top is totally
synchronized (note the flat red stripe). Thus the randomness rapidly disorganizes the
wave front in this case, after which the dynamics settle into the same equilibrium
as before. A multicolor palette facilitates visualization of the interface level sets.
Search postcard: The Stepping Stone Model in One Dimension
 Here we see a spacetime picture of the onedimensional counterpart to the system of
the previous postcard. Each horizontal slice represents the configuration at a certain time;
the initial state is at the top, and the subsequent evolution proceeds down the card.
The dynamics are analogous to those above, except that (i) there are only two neighbors
on the line (left, right), and (ii) the rule is synchronous, meaning that each site
chooses one of two neighbors at random, and then all repainting occurs simultaneously. The
spreading which occurs as we move down the image is onedimensional clustering. Actually all
the cells at the top were colored with random intensities of either red or blue but, due to the
parallel updating, colors interweave to produce a nice collage effect. (Fine checkerboards
such as these are called dithers or dibbles in the graphics world.)
Shelf postcard: The Stepping Stone Model in Two Dimensions
 This process goes back to population geneticist Sewall Wright in
the early 1940's. To make the soup we start from a random configuration,
using a palette of more than 32,000 colors. Then we impose a supremely
simple iterative rule: choose a site at random from the whole array, choose
one of its four nearest neighbors, and give the former cell the color of the
latter. In other words, a random site "eats" a random neighbor. Over
time increasingly large regions of solid color are formed until one
species takes over any finite region. For a nice treatment of this class
of processes, also known as voter models, see Durrett, R.
Lecture Notes on Particle Systems and Percolation. Wadsworth & Brooks/Cole,
Pacific Grove CA, 1988
Sink postcard: A Directed Random Interface from Disorder
 Imagine a row of soldiers marching across a field, or
alternatively a linear array of processors performing some computation
in parallel. Each individual or machine proceeds at the same rate, subject
to random fluctuations, except that anyone ahead of a neighbor must wait
until that neighbor catches up. This simple prescription defines a
cellular automaton directed interface model that exhibits "hydrodynamic"
effects leading to an interesting selforganized equilibrium profile.
When the initial distribution of positions is highly disordered, so that
alignment occurs over time (moving up), we obtain an interesting pseudo3d
effect. For a theoretical discussion see my paper, "SelfOrganization of
Random Cellular Automata: four snapshots." In Probability and Phase
Transition, ed. G. Grimmett. Kluwer, Dordrecht, 1994, 4967.

