Interacting particle system

In probability theory, an interacting particle system (IPS) is a stochastic process ( X ( t ) ) t ∈ R + {displaystyle (X(t))_{tin mathbb {R} ^{+}}} on some configuration space Ω = S G {displaystyle Omega =S^{G}} given by a site space, a countable-infinite graph G {displaystyle G} and a local state space, a compact metric space S {displaystyle S} . More precisely IPS are continuous-time Markov jump processes describing the collective behavior of stochastically interacting components. IPS are the continuous-time analogue of stochastic cellular automata.Among the main examples are the voter model, the contact process, the asymmetric simple exclusion process (ASEP), the Glauber dynamics and in particular the stochastic Ising model. In probability theory, an interacting particle system (IPS) is a stochastic process ( X ( t ) ) t ∈ R + {displaystyle (X(t))_{tin mathbb {R} ^{+}}} on some configuration space Ω = S G {displaystyle Omega =S^{G}} given by a site space, a countable-infinite graph G {displaystyle G} and a local state space, a compact metric space S {displaystyle S} . More precisely IPS are continuous-time Markov jump processes describing the collective behavior of stochastically interacting components. IPS are the continuous-time analogue of stochastic cellular automata.Among the main examples are the voter model, the contact process, the asymmetric simple exclusion process (ASEP), the Glauber dynamics and in particular the stochastic Ising model. IPS are usually defined via their Markov generator giving rise to a unique Markov process using Markov semigroups and the Hille-Yosida theorem. The generator again is given via so-called transition rates c Λ ( η , ξ ) > 0 {displaystyle c_{Lambda }(eta ,xi )>0} where Λ ⊂ G {displaystyle Lambda subset G} is a finite set of sites and η , ξ ∈ Ω {displaystyle eta ,xi in Omega } with η i = ξ i {displaystyle eta _{i}=xi _{i}} for all i ∉ Λ {displaystyle i otin Lambda } . The rates describe exponential waiting times of the process to jump from configuration η {displaystyle eta } into configuration ξ {displaystyle xi } . More generally the transition rates are given in form of a finite measure c Λ ( η , d ξ ) {displaystyle c_{Lambda }(eta ,dxi )} on S Λ {displaystyle S^{Lambda }} . The generator L {displaystyle L} of an IPS has the following form. First, the domain of L {displaystyle L} is a subset of the space of 'observables', that is, the set of real valued continuous functions on the configuration space Ω {displaystyle Omega } . Then for any observable f {displaystyle f} in the domain of L {displaystyle L} , one has L f ( η ) = ∑ Λ ∫ ξ : ξ Λ c = η Λ c c Λ ( η , d ξ ) [ f ( ξ ) − f ( η ) ] {displaystyle Lf(eta )=sum _{Lambda }int _{xi :xi _{Lambda ^{c}}=eta _{Lambda ^{c}}}c_{Lambda }(eta ,dxi )} . For example, for the stochastic Ising model we have G = Z d {displaystyle G=mathbb {Z} ^{d}} , S = { − 1 , + 1 } {displaystyle S={-1,+1}} , c Λ = 0 {displaystyle c_{Lambda }=0} if Λ ≠ { i } {displaystyle Lambda eq {i}} for some i ∈ G {displaystyle iin G} and where η i {displaystyle eta ^{i}} is the configuration equal to η {displaystyle eta } except it is flipped at site i {displaystyle i} . β {displaystyle eta } is a new parameter modeling the inverse temperature. The voter model (usually in continuous time, but there are discrete versions as well) is a process similar to the contact process. In this process η ( x ) {displaystyle eta (x)} is taken to represent a voter's attitude on a particular topic. Voters reconsider their opinions at times distributed according to independent exponential random variables (this gives a Poisson process locally – note that there are in general infinitely many voters so no global Poisson process can be used). At times of reconsideration, a voter chooses one neighbor uniformly from amongst all neighbors and takes that neighbor's opinion. One can generalize the process by allowing the picking of neighbors to be something other than uniform. In the discrete time voter model in one dimension, ξ t ( x ) : Z → { 0 , 1 } {displaystyle xi _{t}(x):mathbb {Z} o {0,1}} represents the state of particle x {displaystyle x} at time t {displaystyle t} . Informally each individual is arranged on a line and can 'see' other individuals that are within a radius, r {displaystyle r} . If more than a certain proportion, θ {displaystyle heta } of these people disagree then the individual changes her attitude, otherwise she keeps it the same. Durrett and Steif (1993) and Steif (1994) show that for large radii there is a critical value θ c {displaystyle heta _{c}} such that if θ > θ c {displaystyle heta > heta _{c}} most individuals never change, and for θ ∈ ( 1 / 2 , θ c ) {displaystyle heta in (1/2, heta _{c})} in the limit most sites agree. (Both of these results assume the probability of ξ 0 ( x ) = 1 {displaystyle xi _{0}(x)=1} is one half.)