This system was introduced by Harris (1974). Again the states are 0 = vacant, 1 = occupied. The birth rates have the same linear form but the death rates are now constant.
(i) An occupied becomes vacant at a rate delta.
(ii) A vacant site becomes occupied at a rate equal to the fraction of the four nearest neighbors that are occupied.
This page is concerned with basic properties of the contact process that can be found in Chapter V of Liggett (1985), in Chapter 4 of Durrett (1988), or in dozens of other sources.
Critical Value for Survival. It is easy to see that if we start with a finite number of occupied sites and delta > 1 then the contact process will die out, i.e., reach the all 0 configuration, with probability 1. We define a critical value delta_c to be the supremum of all of the values of delta so that dying out has a probability < 1.
Critical Value for Equilibrium. There is a second notion of "survival" for the contact process. The contact process is attractive: i.e., increasing the number of 1's increases the birth rate and decreases the death rate.
Theorem 1. If we start an attractive process from all 1's then as t tends to infinity, the distribution of state_t converges to a limit, which is the largest possible stationary distribution.
Of course the limit could assign probability 1 to the all 0 configuration and it will if delta is too large. Let delta_e be the supremum of the values of delta for which the limit is not all 0's.
Duality. For the quadratic contact process we have delta_e > delta_c. However for the contact process these two critical values coincide. Furthermore by using an explicit construction and then working backwards in time much as we did for the voter model one can show:
Theorem 2. Let p_t(A,B) be the probability some site in B is occupied at time t when we start with 1's on A (and 0's elsewhere) at time 0. Then p_t(A,B) = p_t(B,A).
Taking A = all sites and B = a single point we see that the density of occupied sites at time t is the same as surviving until time t starting from a single occupied site.
Math Exercise. (i) Use duality to conclude that if we start with all sites occupied then the probability of having an occupied site in B decreases to a limit. (ii) Use the inclusion-exclusion formula to write any probability involving finitely many sites in terms of the ones in (i) to see that the finite dimensional distributions have a limit as time t tends to infinity.
s3 Exericse. Start the two dimensional contact process from the "Full" initial condition and look at the graph the density of occupied sites versus time. When delta = 0.4 this graph will plateau at about 0.5. Of course there will be flucutations about this level due to the fact that we are observing the density on a finite grid. When delta = 0.6 the limit is about 0.05. Numerical results tell us the critical value of the two dimensional contact process is about 0.607. See Brower, Furman and Moshe (178) and Grassberger and de la Torre (1979), or for a discussion of their results consult Buttel, Cox, and Durrett (1993).
Harris, T.E. (1974) Contact interactions on a lattice. Ann. Prob. 2, 969-988
Brower, R.C., Furman, M.A., and Moshe, M. (1978) Critical exponents for the Reggeon quantum spin model. Phys. Lett. B. 76, 213-219
Grassberger, P. and de la Torre, A. (1979) Reggeon field theory (Schlogl's first model) on a lattice: Monte Carlo calculation of critical behavior. Ann. Phys. 122, 373-396
Liggett, T.M. (1985) Interacting Particle Systems. Springer-Verlag, New York
Durrett, R. (1988) Lecture Notes on Particle Systems and Percolation. Wadsworth Pub. Co. Belmont, CA