Contact process (mathematics)
The contact process is a stochastic process used to model population growth on the set of sites formula_1 of a graph in which occupied sites become vacant at a constant rate, while vacant sites become occupied at a rate proportional to the number of occupied neighboring sites. Therefore, if we denote by formula_2 the proportionality constant, each site remains occupied for a random time period which is exponentially distributed parameter 1 and places descendants at every vacant neighboring site at times of events of a Poisson process parameter formula_2 during this period. All processes are independent of  one another and of the random period of time sites remains occupied. The contact process can also be interpreted as a model for the spread of an infection by thinking of particles as a bacterium spreading over individuals that are positioned at the sites of formula_1, occupied sites correspond to infected individuals, whereas vacant correspond to healthy ones. The main quantity of interest is the number of particles in the process, say formula_5, in the first interpretation, which corresponds to the number of infected sites in the second one. Therefore, the process "survives" whenever the number of particles is positive  for all times, which corresponds to the case that there are always infected individuals in the second one. For any infinite graph formula_1 there exists a positive and finite critical value formula_7 so that if formula_8 then survival of the process starting from a finite number of particles occurs with positive probability, while if formula_9 their extinction is almost certain. Note that by and the infinite monkey theorem, survival of the process is equivalent to formula_10, as formula_11, whereas extinction is equivalent to formula_12, as formula_11, and therefore, it is natural to ask about the rate at which formula_10 when  the process survives. Mathematical Definition. If the state of the process at time formula_15 is formula_16, then a site formula_17 in formula_1 is occupied, say by a particle, if formula_19 and vacant if formula_20. The contact process is a continuous-time Markov process with state space formula_21, where formula_1 is a finite or countable graph, usually formula_23, and a special case of an interacting particle system. More specifically, the dynamics of the basic contact process is defined by the following transition rates: at site formula_17, where the sum is over all the neighbors formula_27 of formula_17 in formula_1. This means  that each site waits an exponential time with the corresponding rate, and then flips (so 0 becomes 1 and 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 formula_30 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 formula_15, formula_5 is approximately equal to formula_33 for some positive constant formula_34. Ted Harris (1974) proved that, if the process survives, then the rate of growth of formula_5 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 formula_15, formula_5 equals formula_38 and the error equals formula_39 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 formula_40 as proved in 2018.