## Long Range Limits

If we consider the contact process on a grid with N sites and modify the
rules so that all sites are neighbors then the number of occupied sites
at time *t* is a Markov chain N(t) with transition rates:

N(t) to N(t)-1 at rate *delta*N(t).

N(t) to N(t)+1 at rate N(t)(1 - N(t)/N).

If we let u_N(t) = N(t)/N be the fraction of occupied sites and let N to
infinity then the u_N (i.e., u subscript N) converges to the solution
of the **"mean field" ordinary differential equation**

The term in quotes refers to the fact that each site only feels the
average value of all the other sites. This equation can also be obtained
directly from the spatial model by letting u(t) be the fraction of
sites occupied at time t and assuming that adjacent sites are
independent. Since the second recipe is simpler we will use it for
computations.

The mean field ODE for the contact process predicts that *delta*_c
is 1 and for *delta* < *delta*_c the equilibrium density
of occupied sites is 1 - *delta*. In the nearest neighbor contact
process there is a significant positve correlation between the
states of neighboring sites (see Harris (1977)) so this overestimates
the critical value.

Bramson, Durrett, and Swindle (1989), have shown:

**Theorem.** If we change
the definition of the spatial model so that two sites *x* and
*y* are neighbors if they differ by at most M in any coordinate,
then for large M, the mean field calculations
of the critical value and equilibrium are almost exact.

**s3 Exercise.** Simulate the model with range is 5, and
*delta* = 0.75 to see that in this case the equilibrium
density is about 13% compared with the limiting value of 25%.

This strategy has been used to facilitate the study of other
models. See the pages on grass bushes trees
and competition of species.

Harris, T.E. (1977) A correlation inequality for Markov processes
in partially ordered state spaces. *Ann. Probab.* **6**,
355-378

Bramson, M,, Durrett, R. and Swindle, S. (1989) Statistical mechanics
of crabgrass. *Ann. Probab.* **17**, 444-481

Swindle, G. (1990) A mean field limit of the contact process with large
range. *Prob. Theory. Rel. Fields.* **88**, 261-282

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