Contact process (mathematics)
   HOME

TheInfoList



OR:

The contact process is a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
used to model population growth on the set of sites S of a
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
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 \lambda the proportionality constant, each site remains occupied for a random time period which is
exponentially distributed In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant averag ...
parameter 1 and places descendants at every vacant neighboring site at times of events of a Poisson process parameter \lambda during this period. All processes are
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independ ...
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 S, 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 N_, 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 S there exists a positive and finite critical value \lambda_c so that if \lambda>\lambda_c then survival of the process starting from a finite number of particles occurs with positive probability, while if \lambda<\lambda_c their extinction is almost certain. Note that by and the
infinite monkey theorem The infinite monkey theorem states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, such as the complete works of William Shakespeare. In fact, the monkey would ...
, survival of the process is equivalent to N_\to\infty, as t\to\infty, whereas extinction is equivalent to N_\to 0, as t\to\infty, and therefore, it is natural to ask about the rate at which N_\to\infty when the process survives.


Mathematical Definition

If the state of the process at time t is \xi_, then a site x in S is occupied, say by a particle, if \xi_(x)=1 and vacant if \xi_(x)=0. The contact process is a continuous-time
Markov process A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happe ...
with state space \^S, where S is a finite or countable
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
, usually \mathbb^d, and a special case of an
interacting particle system In probability theory, an interacting particle system (IPS) is a stochastic process (X(t))_ on some configuration space \Omega= S^G given by a site space, a countable-infinite graph G and a local state space, a compact metric space S . More ...
. More specifically, the dynamics of the basic contact process is defined by the following transition rates: at site x, :1\rightarrow0\quad\text1, :0\rightarrow1\quad\text\lambda\sum_\xi_(y), where the sum is over all the neighbors y of x in S. 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 Percolation (from Latin ''percolare'', "to filter" or "trickle through"), in physics, chemistry and materials science, refers to the movement and filtering of fluids through porous materials. It is described by Darcy's law. Broader applicatio ...

The contact process is a
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
that is closely connected to
percolation theory In statistical physics and mathematics, percolation theory describes the behavior of a network when nodes or links are added. This is a geometric type of phase transition, since at a critical fraction of addition the network of small, disconnected ...
. Ted Harris (1974) noted that the contact process on \mathbb^d 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 \mathbb^ in the direction of increasing coordinate-value.


The

Law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
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 t, N_ is approximately equal to c t for some positive constant c= c(\lambda). Ted Harris (1974) proved that, if the process survives, then the rate of growth of N_ is at most and at least linear in time. A weak
law of large numbers In probability theory, the law of large numbers (LLN) is a theorem that describes the result of performing the same experiment a large number of times. According to the law, the average of the results obtained from a large number of trials shou ...
(that the process
converges in probability In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to ...
) 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 In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to ...
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 In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...

Durrett
conjectured In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
in survey papers and lecture notes during the 80s and early 90s regarding the
central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...
for the
Harris' The Harris Company was a retail corporation, based in San Bernardino, California, that operated a chain of department stores named Harris', all in Southern California. Philip, Arthur, and Herman Harris - nephews of founder Leopold Harris of what ...
contact process, viz. that, if the process survives, then for all large t, N_ equals ct and the error equals \sigma\sqrt t multiplied by a (random) error distributed according to a standard Gaussian distribution.. Durrett's
conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
turned out to be correct for a different value of \sigma 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 Stochastic processes Lattice models