Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model
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Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model D. Coupier October 30, 2006 Laboratoire Paul Painleve, CNRS UMR 8524, Universite Lille 1 E-Mail address : Mail address : Laboratoire Paul Painleve, Universite Lille 1, Cite Scientifique, 59655 Villeneuve d'Ascq Cedex, France. Telephone : 33 (0)3 20 43 67 60 Fax : 33 (0)3 20 43 43 02 Abstract A d-dimensional ferromagnetic Ising model on a lattice torus is consid- ered. As the size of the lattice tends to infinity, two conditions ensuring a Poisson approximation for the distribution of the number of occurrences in the lattice of any given local configuration are suggested. The proof builds on the Stein-Chen method. The rate of the Poisson approximation and the speed of convergence to it are precised and make sense for the model. Thus, the two sufficient conditions are traduced in terms of the magnetic field and the pair potential. In particular, the Poisson approximation holds even if both potentials diverge. Key words : Poisson approximation, Ising model, ferromagnetic interaction, Stein- Chen method. AMS Subject Classification : 60F05, 82B20. 1
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- large magnetic
- local configurations
- given local
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- configuration ?
- µa
- poisson approximation
- positive integer
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Two sufficient conditions for Poisson approximations in the ferromagnetic Ising model
D. Coupier
October 30, 2006
LaboratoirePaulPainlev´e,CNRSUMR8524,Universit´eLille1
E-Mail address :rfel.1c.diipuovadivunil-l@merh.at
Mail address :ePautoirboraLaevinU,e´velniaPl,e1llLi´eitrs Cit´eScientifique,59655Villeneuved’AscqCedex,France. Telephone :33 (0)3 20 43 67 60 Fax :33 (0)3 20 43 43 02
Abstract
Ad-dimensional ferromagnetic Ising model on a lattice torus is consid-ered. As the size of the lattice tends to infinity, two conditions ensuring a Poisson approximation for the distribution of the number of occurrences in the lattice of any given local configuration are suggested. The proof builds on the Stein-Chen method. The rate of the Poisson approximation and the speed of convergence to it are precised and make sense for the model. Thus, the two sufficient conditions are traduced in terms of the magnetic field and the pair potential. In particular, the Poisson approximation holds even if both potentials diverge.
Key words :Poisson approximation, Ising model, ferromagnetic interaction, Stein-Chen method.
AMS Subject Classification :60F05, 82B20.
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Introduction
Suppose{Iα}α∈Γfamily of indicator random variables, with the prop-is a finite erties that the probabilitiesIP(Iα= 1) are small and that there is not too much dependence between theIα’s. Then, the “the law of small numbers” says the sum Pα∈ΓIα The “birthday prob-should have approximately a Poisson distribution. lem” and its variants (see Chen [5] and Janson [14]), the theory of random graphs (seeBollob´as[3]forageneralreferenceorthefamouspaperofErd¨osandR´enyi[8]) and the study of words in long DNA sequences (see for instance Schbath [16]) are examples in which a law of small numbers takes place. As the situation studied in this paper, these examples can be viewed as problems of increasing size (i.e. the car-dinality of Γ tends to infinity) in which the sumPα∈ΓIαhas a Poisson limit. Two methods are often used for proving Poisson approximations; the moment method (see [3] p. 25) and the Stein-Chen method (see Arratia et al. [1], Barbour et al.[2] for a very complete reference, or [5] for the original paper of Chen). The second one offer two main advantages. Only the first two moments need to be computed and a bound of the rate of convergence is obtained. However, the Stein-Chen method requires to restrict our attention to variables which satisfy the FKG inequality [11]. This is the case of spins of a ferromagnetic Ising model. Let us consider a lattice graph in dimensiond≥1, with periodic boundary conditions (lattice torus). The vertex set isVn={0, . . . , n−1}d. The integern will be called thesize edge set, denoted byof the lattice. TheEn, will be specified by defining the set of neighborsV(x) of a given vertexx:
V(x) ={y6=x∈Vn,ky−xkp≤ρ},(1) where the substraction is taken componentwise modulon,k ∙ kpstands for theLp norm inRd(1≤p≤ ∞), andρis a fixed parameter. For instance, the square lattice is obtained forp=ρ the Replacing= 1.L1norm by theL∞norm adds the diagonals. From now on, all operations on vertices will be understood modulo n. In particular, each vertex of the lattice has the same number of neighbors; we denote byVthis number. Aitarnocogunfiis a mapping from the vertex setVnto the state space{−1,+1}. Their set is denoted byXn={−1,+1}Vnand called theconfiguration set. The Ising model is classically defined as follows (see e.g. Georgii [13] and Malyshev and Minlos [15]).
Definition 1.1LetGn= (Vn, En)be the undirected graph structure with finite vertex setVnand edge setEn. Letaandb Thebe two reals.Ising modelwith parametersaandbis the probability measureµa,bonXn={−1,+1}Vndefined by: ∀σ∈ Xn, (σ)Za,bexpax∈XVn{x,y}∈Enσ(x)σ(y),(2) µa,b= 1σ(x) +bX where the normalizing constantZa,bis such thatPσ∈Xnµa,b(σ) = 1 .
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