vice versa). Connection to Percolation The contact process is a stochastic process that is closely connected to percolation theory. Ted Harris (1974) noted that the contact process on when infections and recoveries can occur only in discrete times corresponds to one-step-at-a-time bond percolation on the graph obtained by orienting each edge of in the direction of increasing coordinate-value. The Law of large numbers on the integers A law of large numbers for the number of particles in the process on the integers informally means that for all large , is approximately equal to for some positive constant . Ted Harris (1974) proved that, if the process survives, then the rate of growth of is at most and at least linear in time. A weak law of large numbers (that the process converges in probability) was shown by Durrett (1980). A few years later, Durrett and Griffeath (1983) improved this to a strong law of large numbers, giving almost sure convergence of the process. Die out at criticality For contact process on all integer lattices, a major breakthrough came in 1990 when Bezuidenhout and Grimmett showed that the contact process also dies out almost surely at the critical value. Durrett's conjecture and the central limit theorem Durrett conjectured in survey papers and lecture notes during the 80s and early 90s regarding the central limit theorem for the Harris' contact process, viz. that, if the process survives, then for all large , equals and the error equals multiplied by a (random) error distributed according to a standard Gaussian distribution. Durrett's conjecture turned out to be correct for a different value of as proved in 2018. References C. Bezuidenhout and G. R. Grimmett, The critical contact process dies out, Ann. Probab. 18 (1990), 1462–1482. Durrett, Richard (1988). "Lecture Notes on Particle Systems and Percolation", Wadsworth. Durrett, Richard (1991). "The contact process, 1974–1989." Cornell University, Mathematical Sciences Institute. Thomas M. Liggett, "Stochastic Interacting Systems: Contact, Voter and Exclusion Processes", Springer-Verlag, 1999. Stochastic processes Lattice models