Almost sure quasilocality in the random cluster modelCharles-Edouard P�sterD�epartement de Math�ematiques, Ecole Polytechnique F�ed�erale de LausanneCH-1015 Lausanne SwitzerlandKoen Vande Velde�Instituut voor Theoretische Fysica, K.U. LeuvenCelestijnenlaan 200D, B-3001 Leuven BelgiumAbstractWe investigate the Gibbsianness of the random cluster measures �q;p and ~�q;p, ob-tained as the in�nite-volume limit of �nite volume measures with free and wiredboundary conditions. For q > 1, the measures are not Gibbs measures, but it turnsout that the conditional distribution on one edge, given the con�guration outsidethat edge is almost surely quasilocal.keywords: random cluster model, non-Gibbs states,quasilocality of conditional distributions.

�Aspirant N.F.W.O. Belgium 1

1 Introduction.In recent years, it has become apparent (see [1] and references therein ) that not all statesof physical interest for statistical mechanics are Gibbs measures [2]. Examples can befound in renormalization group theory, where applying renormalization group transfor-mations to Gibbs states may lead out of the class of Gibbs measures [1]. Other examplescome from the theory of interacting particle systems, where nonreversible processes mayhave non-Gibbsian stationary states [3]. A general theory of non-Gibbsian states is notavailable. All one can do for the moment is investigate particular models and try toclassify the examples one has of non-Gibbsian states. One way of classi�cation is thefollowing. Gibbs states verify the property of quasilocality. This means that conditionalexpectations of local events are continuous functions of the con�guration one conditionson. Non-Gibbsian states (if they are not non-Gibbsian for the reason that there are someconstraints or hard-core interactions) lack this quasilocality property. A way of classi�-cation is therefore to look how large the set is on which quasilocality fails [4]. In manycases, this seems very di�cult [5, 4].In this paper we investigate the free and wired random cluster measures with q > 1 and0 < p < 1. We show that we have almost sure quasilocality of conditional distributions,but no quasilocality everywhere. At the same time it makes rigorous the common wisdomthat the random cluster model has nonlocal features.The paper is organized as follows. In section 2 we present the model and state the mainresult of the paper. In section 3 we construct regular conditional distributions for ourmeasures. In section 4 a proof of almost sure quasilocality is given.2 Model and main result.We consider the square lattice 2. The set of edges of 2 is denoted by 2�. The modelunder consideration is the random cluster model. The con�guration space is = f1; 0g 2�.On we put the product topology and its Borel �-�eld F . Con�gurations are denotedby !, � or �. The value of ! at e 2 2� is !e; the restriction of ! to � � 2� is !� := f!e :e 2 �g. To each edge e 2 2� we assign a variable �e with values in f0; 1g, �e(!) := !e;�e(!) = 1 declares the edge open and �e(!) = 0 if the edge is closed. Two sites x; y aresaid to be connected if there is a �nite path via open edges from x to y. A cluster is amaximal set of connected sites. We �x 0 < p < 1 and q > 1. For the construction of thefree boundary condition state, we start with a �nite set of edges � � 2�. We de�ne aprobability measure �� by its weights:�q;p� (�) := 8<: 1Z�pN1(��)(1� p)N0(��)qc(�) if �e = 0 outside �0 otherwise. (2.1)2

Z� is a normalization constant. N1(��) is the number of open edges of � in �; N0(��) isthe number of closed edges of � in �; c(�) is the number of clusters of �.We proceed in a similar way to construct the wired state ~�q;p. The �nite volume measures~�q;p� are de�ned by~�q;p� (�) := 8<: 1Z�pN1(��)(1� p)N0(��)qc(�) if �e = 1 outside �0 otherwise. (2.2)The following results are well known [6].Lemma 1 1. �q;p� and ~�q;p� have the FKG property, i.e. for any increasing functionsg; h on �q;p� (gh) � �q;p� (g)�q;p� (h)~�q;p� (gh) � ~�q;p� (g)~�q;p� (h) (2.3)2. For any increasing function g on ,�q;p� (g) � �q;p�0 (g)~�q;p� (g) � ~�q;p�0 (g); (2.4)if � � �0.3. The weak limits �q;p := lim� �q;p� and ~�q;p := lim� ~�q;p� exist.It is also well known that there exists a critical value pc(q) for p above which thereis percolation in the state �q;p and below which there is absence of percolation. Bypercolation is meant the almost sure existence of an in�nite connected cluster. O� thecritical point, it is believed that �q;p = ~�q;p and that there is thus a unique state for themodel (because free and wired boundary conditions are extremal in FKG sense). It canbe proven rigorously for q = 1; 2; : : : by the existing connection with the Potts model (see[6] for more information on this). Thus �q;p and ~�q;p can only di�er at pc(q), and theyindeed do at high integer values of q where one can make the connection with a �rst ordertransition for the Potts model. For notational convenience we will drop the superscriptq; p unless explicitly needed.De�nition 1 We call a function g on quasilocal at � i� for any � > 0, there exists a�nite region �� such that sup�:���=��� jg(�)� g(�)j < �: (2.5)The following result is well known [7, 8, 1]. 3

Lemma 2 Let F e be the �-algebra of events not depending on !e. The measure � on isa Gibbs measure i� for every edge e there exists a version �e(�ej�) of IE�[�ejF e] satisfying1. 0 < �e(�ej�) < 12. �e(�ej�) is quasilocal at every � 2 .We can now state the main result of the paper.Theorem 1 Let � and ~� be the probability measures constructed above.1. � is not a Gibbs measure. No version of IE�[�ejF e] is quasilocal everywhere.2. There exists a version of IE�[�ejF e] which is quasilocal �-a.s..The same is true for ~�.3 Conditional probabilities.Let us now construct a version �e(�ej�) of the conditional probability IE�[�ejF e]. Let �be a �nite subset of 2� and �c := 2�n�. We denote by 0 the con�guration in whichevery edge is closed, and by 1 the con�guration in which every edge is open. We de�ne�e(�ej�) for all � = ��0�c as the conditional expectation of �e with respect to ��; sincethose con�gurations are dense in we then extend the de�nition by a limiting procedure.(The proof of the existence of the limit is given after the next Lemma.) Thus,�e(�ej��0�c) := ��(�ej��nfeg0�c) (3.1)�e(�ej�) := lim� �e(�ej��0�c): (3.2)Similarly we construct a version ~�e(�ej�) of IE~�[�ejF e].~�e(�ej��1�c) := ~��(�ej��nfeg1�c) (3.3)~�e(�ej�) := lim� ~�e(�ej��1�c): (3.4)Lemma 3 Let V 3 e be a �nite subset of 2�. For a F e-measurable function g�V (�e(�ej�)g) = �V (�eg) (3.5)~�V (~�e(�ej�)g) = ~�V (�eg) (3.6)4

Proof: �V (�e(�ej�)g) = Z �V (d!)�e(�ej!)g(!)= Z �V (d!)�e(�ej!V 0V c)g(!V 0V c)= Z �V (d!)IE�V [�ejF e]g(!)= �V (�eg)The proof of the second statement is similar.Let us compute now more explicitly our expression for �e(�ej�). Denote by �0 the con�g-uration equal to � o� e and equal to 0 on e and by �1 the con�guration equal to � o� eand equal to 1 on e. Now �e(�ej��0�c) = ��(�1)��(�1) + ��(�0) : (3.7)It is thus clear that�e(�ej��0�c) = 8><>: pp+q(1�p) if c(�0�0�c) = c(�1�0�c) + 1p if c(�0�0�c) = c(�1�0�c): (3.8)The event where c(�0�0�c) = c(�1�0�c)� 1 is impossible. Indeed, since isolated lattice sitesare also counted as clusters, the creation of an open edge cannot increase the number ofclusters.For any �nite � we say that two sites x and y are connected inside � for the con�guration� if they are connected for the con�guration ��0�c . Similarly, we say that two points areconnected outside � for � if they are connected for ��1�c. Let e be the edge with sites xand y as endpoints. De�ne the following events:Eiu;� := fx is connected to y inside �g;and Eiu := [� Eiu;�:Similarly, de�ne Eou;� := fx is connected to y outside �g;and Eou := [� Eiu;�![ \� Eou;�! :5

Let �iu denote the indicator function of Eiu. Denote with �ou the indicator function of Eou.Then c(�0�0�c) = c(�1�0�c) + 1 i� �0 62 Eiu;�c(�0�0�c) = c(�1�0�c) i� �0 2 Eiu;�:Now we want to take the limit �%1. If �0 2 Eiu there has to exist some �nite volume� such that �0 2 Eiu;�. Therefore, for each such � the limit over � exists and�e(�ej�) = pp+ q(1� p) �1� �iu(�0)�+ p�iu(�0)= pp+ q(1� p) + p� pp+ q(1� p)!�iu(�0): (3.9)It is clear that �e(�ej�) is a nonlocal, nonnegative and increasing function since q > 1. Ina similar way ~�e(�ej��1�c) = 8><>: pp+q(1�p) if c(�0�1�c) = c(�1�1�c) + 1p if c(�0�1�c) = c(�1�1�c): (3.10)It is again clear then thatc(�0�1�c) = c(�1�1�c) + 1 i� �0 62 Eou;�c(�0�1�c) = c(�1�1�c) i� �0 2 Eou;�:For �0 2 Eiu we already know that we can take the limit �%1. Thus now take � 2 ,�0 2 Eou but �0 62 Eiu. This � is such that �0 2 Eou;� for all �nite �. Thus,~�e(�ej��1�c) = p:The limit over � thus exists and equals p. It is then clear that~�e(�ej�) = pp+ q(1� p) + p� pp+ q(1� p)!�ou(�0): (3.11)The expressions for �e(�ej�) and ~�e(�ej�) di�er thus on the set of con�gurations � suchthat in �0 both x and y are connected to in�nity, but not to each other.The following Lemma is an adaptation of Lemma 3.1 in [4].6

Lemma 4 Let g be a monotone bounded function. Ifg(!) = lim� g(!�0�c) (3.12)then �(g) = lim� ��(g): (3.13)If g(!) = lim� g(!�1�c) (3.14)then ~�(g) = lim� ~��(g): (3.15)Proof:We just prove (3.13) for monotone increasing functions. The proof of the other statementsis similar. We denote g�(!) := g(!�0�c). Since g is increasing, g� � g�0 for � � �0. LetM � � � N , jN j <1. ��(gM) � ��(g�) = ��(g) � �N(g�): (3.16)The left inequality is due to the above remark, the right one is due to Lemma 1 since g�is monotone increasing. Since g� is a local function, we can take the limit over N , andget (Lemma 1) ��(gM) � ��(g) � supN �N(g�) = �(g�): (3.17)By hypothesis lim� g� = g; by the monotone convergence theorem we obtain�(gM) � lim� inf ��(g) � lim� sup��(g) � �(g): (3.18)Finally taking the limit over M yields�(g) � lim� ��(g) � �(g): (3.19)Proposition 1 �e(�ej�) is a version of IE�[�ejF e].Proof: It su�ces to prove that for any nonnegative increasing localF e-measurable functiong �(�eg) = �(�e(�ej�)g): (3.20)7

Because both �e and g are local we have from weak convergence and Lemma 3 that�(�eg) = lim� ��(�eg) (3.21)= lim� ��(�e(�ej�)g):Now �e(�ej�) is nonnegative and increasing and so is g. Then so is �e(�ej�)g. By de�nition�e(�ej!) = lim� �e(�ej!�0�c); since g is local the same is true for �e(�ej�)g. We can there-fore apply Lemma 4 which guarantees the convergence of ��(�e(�ej�)g) to �(�e(�ej�)g).This concludes the proof.In the same manner we prove that ~�e(�ej�) is a version of IE~�[�ejF e].4 Almost sure quasilocality.We now investigate the quasilocality properties of �e(�ej�). We follow [1] section 4.5.3.Because �e(�ej�) is only one version of IE�[�ejF e], to prove non-quasilocality of IE�[�ejF e],we have to show that no function that equals �e(�ej�) �-a.s. is quasilocal everywhere. Wesay then that �e(�ej�) displays an essential non-quasilocality. Because � gives nonzeroprobability to any open set, it su�ces to investigate the function �e(�ej�) on a neighbor-hood of �, to search for essential non-quasilocality of �e(�ej�) at �. A neighborhood of �is constructed in the following way. Fix a �nite set �, thenN�(�) := f� 2 : �� = ��g (4.1)is a neighborhood of �.Proposition 2 No version of IE�[�ejF e] is quasilocal everywhere.Proof:The proof is given in [1]. We repeat it here for the sake of completeness and because itgives more insight in the properties of �e(�ej�).Let �n := [�n; n]2. Let � be the con�guration which sets �f := 1 on parallel rays runningfrom x and y to in�nity, perpendicular to the edge e, and sets �f = 0 on all other edges.We choose two subsets of N�n(�): N 1�n(�) in which an open edge in �n+1n�n connectsthe two rays, N 0�n(�) in which all edges of �n+1n�n are closed, so the parallel rays cannotbe connected no matter what the con�guration ouside �n+1 is. Now for all �1;n 2 N 1�n(�),�2;n 2 N 0�n(�) �e(�ej�1;n)� �e(�ej�2;n) = p� pp+ q(1� p) > 0; (4.2)uniformly in n.Since N 1�n(�) and N 0�n(�) carry positive �-measure, it follows that no function that equals8

�e(�ej�) �-a.s. can be quasilocal at �.We set out now to �nd how large the set of con�gurations is, where �e(�ej�) exhibitsessential non-quasilocality. It turns out that this set is rather small in measure-theoreticalsense.Lemma 5 �-a.s. there exists no more than one in�nite cluster. The same holds for ~�.Proof:As a consequence of the Burton-Keane uniqueness Theorem [9], the result follows fromtranslation invariance of � and the so-called �nite-energy property, i.e.0 < IE�[�ejF e] < 1; �-a.s.: (4.3)But this condition is easily veri�ed from expression (3.8) for �e(�ej�) and Proposition 1.The same reasoning applies to ~�.Proposition 3 The function �e(�ej�) is quasilocal �-a.s..Proof:According to Lemma 5, the set 1 of con�gurations where there is no in�nite cluster ora unique in�nite cluster carries full measure. Take thus any � 2 1. Suppose �rst that �has no in�nite cluster. Then there exists some �nite set �, e 2 �, for which no site in �is connected to �c. But then �e(�ej�) = �e(�ej�) whenever � and � agree inside �. Thisproves locality of �e(�ej�) at such �.Now take any con�guration � 2 1 that has a unique in�nite cluster. If not both x and yare connected to in�nity, then again there exists a �nite set � such that �e(�ej�) = �e(�ej�)whenever � and � agree inside �. So suppose that both x and y are connected to in�n-ity. Because of the uniqueness of the in�nite cluster, there exists now a �nite set �such that x and y are connected by a path of open edges within �. In that case again�e(�ej�) = �e(�ej�) for all � that agree with � inside �. Hence �e(�ej�) is quasilocal forall con�gurations � 2 1.Because both � and ~� have no more than one in�nite cluster it is now clear from ex-pressions (3.9) and (3.11) that �-a.s. and ~�-a.s. �e(�ej�) = ~�e(�ej�). It follows that�e(�ej�) is a version of both IE�[�ejF e] and IE~�[�ejF e] and that �e(�ej�) is also quasilocal~�-a.s..Acknowledgment: KVV thanks the EPFL in Lausanne, where part of the work was done,for kind hospitality. 9

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