This is perhaps the simplest possible model. Each site in the square lattice is either occupied (in state 1) or vacant (in state 0).

(i) A site once occupied stays occupied forever.

(ii) A vacant site becomes occupied at a rate equal to the fraction of the four nearest neighbors that are occupied.

If we start with only the origin occupied at time 0 then each site in lattice will eventually become occupied (and remain occupied forever). Things get interesting when we we let B(t) be the set of lattice points occupied at time t ask

** At what rate does the blob B(t) grow? **

Eden (1961) was the first to ask this question, but the solution remained elusive until Richardson (1973) showed

**Theorem.** B(t)/t has
a limiting shape, which is roughly but not exactly circular.

**s3 Exercise.** Check the last theorem by
running the model starting from a single occupied
site. (Choose the initial condition "rectangle" and set the height and width
equal to 1.)

**Sketch of Proof.** The first step in Richardson's asymptotic
shape result is to show that if t(n,0) is the first time the point
(n,0) is occupied then t(n,0)/n converges to a constant c(1,0).
This is accomplished by using the **subadditive ergodic theorem**.
For details, see e.g., Smythe and Wierman (1977), Cox and Durrett (1981),
or Kesten (1986).

** Research Problem. **
Having a law of large numbers for t(n,0) it is natural to ask if there is
a central limit theorem. Before you say "of course", we would like to
note that simulations (Zabolitsky and Stauffer (1986), Wolf and
Kertesz (1987)) suggest the standard deviation of t(n,0)
is of order n^{1/3}, i.e., *n* to the one third power, instead
of the usual square root of *n*. There are some heuristic
arguments in support of this result. See e.g., Kardar, Parisi,
and Zhang (1986). However, all that is known rigorously
is that the fluctuations are no worse than square root of *n*.
See Kesten (1993) and Alexander (1993), and the related work reported
on in Newman (1994).

Eden, M. (1961) A two dimensional growth process. * Proc. 4th
Berkeley Symp.*, IV, 222-239

Richardson, D. (1973) Random growth in a tesselation. * Proc. Camb.
Phil. Soc.* **74**, 515-528

Smythe, R. and Weirman, J.C. (1977) *First Passage Percolation
On The Square Lattice.* Springer Lecture Notes in Math 671

Cox, J.T. and Durrett, R. (1981) Some limit theorems for first
passage percolation with necessary and sufficient conditions.
*Ann. Prob.* **9**, 583-603

Kesten, H. (1986) Aspects of First Passage Percolation. In
*Springer Lecture Notes in Math 1180*

Kardar, M., Parisi, G., and Zhang, Y.C. (1986) Dynamic scaling
fo growing interfaces. *Phys. Rev. Letters.* **56**, 889-892

Zabolitsky, J.G. and Stauffer, D. (1986) Simulation of large
Eden clusters. *Phys. Rev. A* **34**, 1523-1530

Wolf, D. and Kertesz, J. (1987) Noise reduction in Eden models.
*J. Phys. A* **20**, L257-L261

Kesten, H. (1993) On the speed of convergence in first passage
percolation. *Ann. Appl. Prob.* **3**, 296-338

Alexander, K. (1993) A note on some rates of convergence in
first-passage percolation. *Ann. Appl. Prob.* **3**, 81-90

Newman, C. (1994) A surface view of first passage percolation.
*Proc. Int. Congress of Math, Zurich.*