This is sometimes called Schlogl's second model. See Schlogl (1972) and Grassberger (1982). Again 0 = vacant and 1 = occupied, but now it takes two particles to make a new one.

(i) An occupied becomes vacant at a rate *delta*.

(ii) A vacant site becomes occupied at a rate equal to k/4 where k is number of diagonally adjacent pairs of occupied neighbors.

The ** Critical Value for Survival ** is 0. If the initial
configuration starts inside a rectangle it can never give birth outside
of the rectangle and hence is doomed to die out whenever *delta* is
positive.

Somewhat surprisingly, the ** Critical
Value for Equilibrium** is positive. Bramson and Gray (1991)
showed that if *delta* is small enough then the limit starting from
all 1's is a nontrivial stationary distribution.

**s3 Exercises.** To get a feeling for the critical value for
survival set *delta* = 0.075 and start the system from a
25 by 25 rectangle of 0's in a sea of 1's and watch it fill in
the hole. When *delta* = 0.125 the hole grows and the system
dies out. When *delta* = 0.1 things go well for a while
then ...

When *delta* < *delta*_c the system does well
when the density of 1's is large enough. However,
it always has trouble when the density is small.
H.N. Chen (1992) has shown that if we start from product measure
with a small density *p* (i.e., sites are independently occupied with
probability *p*) then P( state_t[x] = 1) tends to 0 for all *x*.

**s3 Exercise.**
To see this in a simulation set *delta* = 0.05 which is
well below the critical value. When p = 0.2 the system dies out,
but when p = 0.3 it approaches equilbrium. In between at p = 0.25
the pictures are the most interesting.

Schlogl, F. (1972) Chemical reaction models for non-equilibrium phase
transitions. *Z. Physik* **253**, 147--161

Grassberger, P. (1982) On phase transitions in Schlogl's second model.
*Z. Phys. B.* **47**, 365--376

Bramson, M. and Gray, L. (1991) A useful renormalization argument.
* Random Walks, Brownian Motion, and Intracting Brownian Motion.*
Edited by R. Durett and H. Kesten. Birkhauser, Boston.

Chen, H.N. (1992) On the stability of a population growth model with
sexual reproduction on Z^2. *Ann. Probab.* **20**, 232-285