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 deltaN(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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