##
**Stochastic interacting systems: contact, voter and exclusion processes.**
*(English)*
Zbl 0949.60006

Grundlehren der Mathematischen Wissenschaften. 324. Berlin: Springer. xii, 332 p. (1999).

The author [“Interacting particle systems” (1985; Zbl 0559.60078)] presented the first comprehensive account of a then rather new field of probability. This book has ever since been the standard reference and basic introduction for everyone entering and working in this field. Now, fifteen years later, with at least three major textbooks and hundreds of research articles added to the field, the author presents his second book which, as he says, is neither a second edition nor a second volume, but rather a review of the developments in some selected subfields of the author’s choice. Indeed, the scope of the book is limited to just three models: the contact process, the voter model, and the exclusion process, to each of which one part of the book is devoted.

The contact process is a continuous time Markov process \(\eta_t\) with state space \(\{0,1\}\), where \(S\) is a connected graph; in the examples treated here \(S\) is either a lattice \(Z^d\) or a homogeneous tree \(T_d\). The process is described as the spreading of the “infected” set \(A\subset S\); each site \(x\in A\) is removed with rate one and any site \(y\) at distance one from \(A\) is added with rate \(\lambda\). The main question is whether the process starting with a single infected site \(x\) reaches the stable state \(A=\emptyset\), or not. In the latter case one says that the process survives; if moreover \(x\) is getting re-infected an infinite number of times, the process is said to survive strongly. These properties depend on the parameter \(\lambda\), and the basic question is whether for a given graph \(S\) varying the parameter will allow to realize all these possibilities. These basic questions are answered for \(Z^d\) and for homogeneous trees. An important question is to relate this behaviour of infinite systems to relevant properties of finite systems. Such systems are studied in Section 3, where it is shown that these distinct behaviours manifest themselves in very different volume dependence of extinction times in the finite system.

Voter models are introduced as continuous time Markov chains on \(\{0,1\}^{\mathbb Z^d}\) whose transition rates are stationary and invariant under global exchange \(0\leftrightarrow 1\), leave the constant configurations invariant, and are attractive. The main issue here is the possibility of coexistence versus clustering: will both \(0\)’s and ones persist for all times, or are they clustering so that eventually one “opinion” will dominate over arbitrarily large regions of space? These questions are addressed in threshold voter models with finite range (i.e. when the probability of changing the opinion at site \(i\) depends only on the opinions on some finite neighborhood of \(i\), and a change of opinion occurs with rate one if in that neighborhood a given opinion dominates by some threshold amount \(T\)).

The final Part III deals with exclusion processes. Here particles places on some countable set perform random walk according to some transition rates \(p(x,y)\), subject to the constraint that no site may be occupied by more than one particle. Essentially only the case on \(\mathbb{Z}^d\) with translationally invariant rates is considered. An important distinction is between symmetric and asymmetric situations; totally asymmetric exclusion processes on \(\mathbb{Z}\) are heavily studied, where particles may walk only in one direction. An important concept is the study of first and second class particles. The former are distinguished by the fact that if they try to go to a site occupied by a second class particle, they do so and exchange their position with the second class one. Sections 2 and 3 of this part are dealing with asymmetric nearest neighbor processes in one dimension. Interesting features here are the appearance of shocks in a process starting from an inhomogeneous initial condition. Section 2 investigates those using coupling techniques while Section 3 introduces an algebraic approach that allows exact computations. Section 4 then treats more general asymmetric processes; a central issue here is a central limit theorem for the motion of a tagged particle.

Each section begins with a general introduction, including of what is to come, and ends with notes and references, making the book both very accessible and comprehensive. An extensive bibliography completes the book which undoubtedly will take a prominent place at the side of its predecessor.

The contact process is a continuous time Markov process \(\eta_t\) with state space \(\{0,1\}\), where \(S\) is a connected graph; in the examples treated here \(S\) is either a lattice \(Z^d\) or a homogeneous tree \(T_d\). The process is described as the spreading of the “infected” set \(A\subset S\); each site \(x\in A\) is removed with rate one and any site \(y\) at distance one from \(A\) is added with rate \(\lambda\). The main question is whether the process starting with a single infected site \(x\) reaches the stable state \(A=\emptyset\), or not. In the latter case one says that the process survives; if moreover \(x\) is getting re-infected an infinite number of times, the process is said to survive strongly. These properties depend on the parameter \(\lambda\), and the basic question is whether for a given graph \(S\) varying the parameter will allow to realize all these possibilities. These basic questions are answered for \(Z^d\) and for homogeneous trees. An important question is to relate this behaviour of infinite systems to relevant properties of finite systems. Such systems are studied in Section 3, where it is shown that these distinct behaviours manifest themselves in very different volume dependence of extinction times in the finite system.

Voter models are introduced as continuous time Markov chains on \(\{0,1\}^{\mathbb Z^d}\) whose transition rates are stationary and invariant under global exchange \(0\leftrightarrow 1\), leave the constant configurations invariant, and are attractive. The main issue here is the possibility of coexistence versus clustering: will both \(0\)’s and ones persist for all times, or are they clustering so that eventually one “opinion” will dominate over arbitrarily large regions of space? These questions are addressed in threshold voter models with finite range (i.e. when the probability of changing the opinion at site \(i\) depends only on the opinions on some finite neighborhood of \(i\), and a change of opinion occurs with rate one if in that neighborhood a given opinion dominates by some threshold amount \(T\)).

The final Part III deals with exclusion processes. Here particles places on some countable set perform random walk according to some transition rates \(p(x,y)\), subject to the constraint that no site may be occupied by more than one particle. Essentially only the case on \(\mathbb{Z}^d\) with translationally invariant rates is considered. An important distinction is between symmetric and asymmetric situations; totally asymmetric exclusion processes on \(\mathbb{Z}\) are heavily studied, where particles may walk only in one direction. An important concept is the study of first and second class particles. The former are distinguished by the fact that if they try to go to a site occupied by a second class particle, they do so and exchange their position with the second class one. Sections 2 and 3 of this part are dealing with asymmetric nearest neighbor processes in one dimension. Interesting features here are the appearance of shocks in a process starting from an inhomogeneous initial condition. Section 2 investigates those using coupling techniques while Section 3 introduces an algebraic approach that allows exact computations. Section 4 then treats more general asymmetric processes; a central issue here is a central limit theorem for the motion of a tagged particle.

Each section begins with a general introduction, including of what is to come, and ends with notes and references, making the book both very accessible and comprehensive. An extensive bibliography completes the book which undoubtedly will take a prominent place at the side of its predecessor.

Reviewer: A.Bovier (Berlin)

### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82B21 | Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics |

82C22 | Interacting particle systems in time-dependent statistical mechanics |

91B12 | Voting theory |