Bond Percolation in Frustrated Systems

The Annals of Probability 1999, Vol. 27, No. 4, 1781–1808 BOND PERCOLATION IN FRUSTRATED SYSTEMS By E. De Santis and A. Gandolfi Università di Roma La Sopienza and Tor Vergate We study occurrence and properties of percolation of occupied bonds in systems with random interactions and, hence, frustration. We develop a general argument, somewhat like Peierls’ argument, by which we show that in Zd , d ≥ 2, percolation occurs for all possible interactions (provided they are bounded away from zero) if the parameter p ∈ 0 1, regulating the density of occupied bonds, is high enough. If the interactions are i.i.d. random variables then we determine bounds on the values of p for which percolation occurs for all, almost all but not all, almost none but some, or none of the interactions. Motivations of this work come from the rigorous analysis of phase transitions in frustrated statistical mechanics systems. 1. Introduction. In this paper we study occurrence and properties of percolation of occupied bonds in models, defined on regular lattices (mostly
d for simplicity), in which percolation is made more difficult by frustration, which is the presence of particular circuits which cannot be fully occupied. This is determined by restrictions given in terms of a preassigned selection J of interactions, that is, real values (both positive and negative in the interesting cases), one for each bond; the circuits for which the product of the interactions is negative are called frustrated and cannot be fully occupied. We study probability measures on bond occupation variables satisfying this constraint. Two closely related classes of such probability measures, both dependent on one parameter, p ∈ 0 1, are introduced: a simpler one, which is just a Bernoulli measure conditioned to avoid full occupation of frustrated circuits, and a more interesting one, which is the FK random cluster model (see Section 2). Percolation, that is, the existence, with positive probability with respect to one of the above-mentioned measures, of an infinite connected chain of occupied bonds is then hampered by the need to avoid full occupation of frustrated circuits. Nonetheless, percolation occurs for p sufficiently close to one, for example, p > 8/9 in the conditionally unfrustrated Bernoulli measure and p > 16/17 in the FK random cluster model, for any J ∈ −1 1 (see Theorems 2.2 and 2.8 below) and in any dimension d ≥ 2. Occurrence of percolation in such models was not at all clear in dimension 2 even for a set of interactions J having probability 1 according to the Bernoulli distribution (see [13] for the two-dimensional FK random cluster model). The formulation of the models above is motivated by the role played by the FK random cluster model as a representation of spin systems, in particular of frustrated spin systems like spin glasses. In the ferromagnetic Ising model at zero external field and inverse temperature β, spontaneous magnetization, Received May 1997; revised April 1999. AMS 1991 subject classifications. Primary 60K35; secondary 82A25, 82A57, 82A68. Key words and phrases. Percolation, frustration, Peierls’ argument, spin glasses. 1781 1782 E. DE SANTIS AND A. GANDOLFI and so phase transition, occurs if and only if percolation of occupied bonds occurs in the related FK random cluster model with parameter p = 1 − e−2β (see [10]), but in the spin glass models [8] this correspondence is not as direct (see [23]). In the Edwards–Anderson (EA) spin glass model (see [8]) on the vertices of Zd , for instance, interactions are randomly and independently chosen with some distribution ℘ symmetric around the origin, and the question is whether phase transition occurs for ℘ -almost all J. It is expected (but not proved) that for low enough temperature, that is, high enough β, phase transition occurs in dimension d ≥ 3 and it is an open question what to expect in dimension d = 2. It was proved in [13] that percolation of occupied bonds occurs in the FK random cluster model in d ≥ 3 for ℘ -almost all J (for at least some simple distribution ℘ ), and this seemed an indication that the picture outlined above could hold. Here, however, we prove that a similar transition to percolation occurs in dimension d = 2 also, so that percolation gives no clear indication on whether a phase transition could take place in the EA model (on the relation between percolation and phase transition for spin glasses see also [12]). On the other hand, it is possible to modify the proof of our main result to show that a ferromagnetic phase transition does take place at low density of antiferromagnetic interactions, obtaining a different proof of some of the results in [6]. These being the motivations, we now restrict our attention to percolation of occupied bonds. Our method in Section 2 shows that for p large enough percolation of occupied bonds occurs, even subject to avoiding full occupation of frustrated circuits, for all J. We think that this method is, by itself, quite general, somewhat like Peierls’ argument, and we think that it might be applicable to discuss percolation of a variety of possibly frustrated systems; in particular, since the argument applies to all interactions J, it can be used in models where a specific nonrandom form of the J is assumed. To illustrate the method, we apply it to the simplest model in Theorem 2.2 and leave further refinements to later parts of the paper. Given a Bernoulli distribution ℘ on the interactions J, a new phenomenon is described in Section 3: for some values of p, percolation might occur for ℘ -almost none but some, or for ℘ -almost all but not all interactions J (these two possibilities, together with the presence or absence of percolation for all J at some other values of p, are shown to occur in Theorem 4.2 for the triangular lattice T). Theorem 3.2 shows that also in the more interesting twodimensional FK random cluster model (with two-valued J) there are regions of p for which percolation occurs for ℘ -almost none but some J. Questions about statements for all versus almost all realizations, as function of some parameter, have been dealt with in a variety of contexts concerning spatial random systems: random coverings [27], Brownian motion [20], uniqueness of percolation cluster [3] and occurrence of percolation under some evolution [16]. In analogy with some of the results of these papers, we find nontrivial regions which could be called critical, that is, in which not all interactions J behave the same. Other results concerning presence or absence of percolation for not necessarily two-valued distributions are discussed in Section 4. BOND PERCOLATION IN FRUSTRATED SYSTEMS 1783 There remain various other issues which are not discussed in the paper. For example, the determination of the influence of the type of transition that we have found on the thermodynamical properties of spin glass systems, in terms for instance of the change in magnetization or correlation, or, more specifically, about percolation of unfrustrated bonds, the determination (of at least the existence) of the critical value for percolation for all interactions J and of the ℘ -dependent critical values for percolation for ℘ -almost all J. For further discussion on the regions in which percolation occurs, see [12]. 2. Percolation for all interactions. The results presented in this section are quite general and can be applied to a variety of models; some of these applications are exploited in the following sections. Here we present the main ideas in their simplest form, referring to a very simple model which basically just captures the idea of (randomized) frustration. First we need some geometrical definitions, which are given directly in Zd for simplicity. Let · indicate the Euclidean distance (we use the same symbol later to indicate cardinality of sets, but no confusion should arise), and consider Zd ⊂ Rd as a graph with set of vertices VZd  = Zd and set of nearest neighbour bonds BZd  = i j  i j ∈ Zd  i − j = 1 . For a bond b = i j , the vertices i and j are called end points; for a set E of bonds, VertE ⊆ Zd is the set of vertices which are end points of at least one bond of E. Also, the origin of Zd is denoted by O. We define a path to be an ordered sequence π = i0  b1  i1      in−1  bn  in  of bonds bm ∈ BZd , m = 1     n and of vertices im ∈ Zd , m = 0     n, such that bm = im−1  im for m = 1     n; the bonds bi are then said to belong to the path and sometimes a path is indicated by the list of its bonds only. Note that with our definition, both bonds and vertices can appear more than once; however, if Vertbm  ∩ Vertbl  = 0 when m − l =  1, then we say that the path is self-avoiding or, briefly, s.a.; note that sometimes the difference between self-avoiding or not might be essential, and we are careful in indicating when a path needs to be or must be self-avoiding. Given two paths π1 = b1      bn  and π2 = c1      cm , we indicate by π1  π2  the sequence b1      bn  c1      cn , and if such sequence, completed by an appropriate selection of vertices, is a path we call it the concatenation of the two paths. Next, we define a circuit to be a path π = i0  b1  i1      in−1  bn  in  such that also i0 = in ; a circuit π = b1      bn  is s.a. if the path b1      bn−1  is such. Finally, define a box to be a set of bonds  such that Vert( = k1  h1  × k2  h2  ×    × kd  hd  ∩ Zd for some ki , hi ∈ Z, ki < hi . In particular, let k be such that Vertk  = −k kd ∩ Zd ; for a box  let ∂ be the boundary of , that is, the bonds of , one end point of which belongs also to a bond not in . Later, we consider sequences of random variables indexed by boxes and we always take the limit in the sense of Van Hove, that is, along sequences of boxes nk such that limk→∞ ∂nk / nk = 0. Next, we discuss configurations of interactions which, until Section 4, are taken two-valued. Let J = Ji j i j∈Zd i−j =1 ∈ −1 1 BZd  = 1784 E. DE SANTIS AND A. GANDOLFI be a given prescription of the interaction between vertices of Zd . We say that a set E of bonds is J-frustrated, or is frustrated with respect to J, if and only if there is a circuit π ⊆ E such that b∈π Jb = −1; we also call such a circuit frustrated. If no frustrated circuit can be found, then E is J-unfrustrated or is unfrustrated with respect to J. A set of vertices V is then frustrated with respect to some J if and only if the set of bonds i j  i j ∈ V is J-frustrated. Third, we identify subsets of BZd  by configurations η = ηij i j∈Zd  i−j =1 ∈ d 0 1 BZ  = H, which we then call frustrated if there exists a circuit π ⊆ η−1 1 such that b∈π Jb = −1; here η−1 1 indicates the set of bonds in which the configuration η takes value +1 (later, by a slight abuse of notation, it also indicates its cardinality). Given a configuration η and a set A ⊂ BZd , we indicate by ηA the restriction of η to A and by η\A the restriction of η to BZd  \ A; sometimes, if A ⊂ B, we use η\A = ηB\A when no confusion can arise. We also identify the restriction ηA of a configuration η with the cylinder of configurations coinciding with ηA on A. We say that a configuration η is J-unfrustrated in a set of bonds S if it is unfrustrated with respect to the restriction of J to η−1 1; hence there is no circuit π ⊂ S ∩ η−1 1 such that b∈π Jb = −1. Given two sets A, B ⊂ BZd , with A ∩ B = ⵰, and two configurations ηA and ηB , we denote by ηA ∨ ηB the configuration ηA∪B which coincides with ηA in A and with ηB in B; the same notation is used with J replacing η. Next, we give the definitions concerning percolation (see also [14]). We call a bond b occupied (in η) if ηb = 1, and a path π occupied if all of its bonds b ∈ π are occupied. Given η, we consider two vertices connected if there is an occupied path containing both vertices; then the set Vertη−1 1 falls apart into maximal connected components called clusters. We say that percolation from the origin (or from a vertex v) occurs if Vertη−1 1 contains at least one infinite cluster containing the origin (or the vertex v, respectively). On the other hand, we say that percolation occurs if Vertη−1 1 contains at least one infinite cluster. Equivalent definitions are obtained as follows. Percolation from the origin is equivalent to the existence of an infinite occupied self-avoiding path π containing the origin. Next, let Cv ∂ be the event that there is an occupied path π whose set of vertices Vertπ includes the vertex v ∈ Vert and a vertex w ∈ Vert∂. Denote by Cv∞ the existence of an infinite cluster including v, then Cv ∞ = ∩k≥k0 Cv ∂k for some k0 . The sets of occupied bonds are described by a random mechanism, a simple form of which is obtained by defining, on the σ-algebra of H generated by cylinders, the following Bernoulli probability measures conditioned to avoiding frustration. The motivation for this definition is that it captures relevant features of the more interesting FK random cluster measures (to be discussed later). Let J ∈  and p ∈ 0 1 and let  = k ⊂ BZd  be a box. Given η = 0 1  ∈ H , let −1 21 νJ p  η  = pη 1 −1 1 − pη 0 UJ η   ZJ p  BOND PERCOLATION IN FRUSTRATED SYSTEMS 1785 where UJ η  is the indicator function that η is unfrustrated with respect to J, that is,
1 if the set η−1  1 is unfrustrated with respect to J, UJ η  = 0 otherwise; ZJ p  is a normalizing factor, and η−1  i, i = 0 1, indicates, with a slight abuse of notation, the cardinality of the set η−1  i on which η takes the value i. Let νJ p be any weak limit of the sequence of probability measures νJ p k taken as k diverges to infinity, where k is a sequence of boxes such that  d  k k = Z , and the limit is taken in the sense of Van Hove (as discussed above). We call every such measure a conditionally J-unfrustrated Bernoulli measure. We are interested in percolation under a measure νJ p . There is no immediate equivalence, in this case, between occurrence of percolation from the origin with positive νJ p probability and occurrence of percolation with νJ p -probability 1; therefore, we focus on the second type of phenomenon for the time being; a simple argument in Lemma 2.7 below shows that percolation a.e. implies percolation from the origin with νJ p positive probability. To simplify the discussion we introduce two critical points, 22 p c d = supp ∀ J ∀ νJ p νJ p percolation occurs = 0 and 23 pc d = inf p ∀ J ∀ νJ p νJ p percolation occurs > 0  This is just one of the possible definitions of critical points (see [24], page 39, for a related discussion and other possible definitions). We remark that, although we cannot determine, in general, monotonicity properties in p of νJ p percolation occurs for fixed J, such monotonicity can be shown for the configuration of interactions J such that J ≡ 1. In fact, in this case any measure νJ p  , and thus any weak limit νJ p , is just a Bernoulli measure with occupation density p; monotonicity in p of the probability that percolation occurs follows from the FKG inequality. This, in particular, implies that p c d ≤ pc d. A standard technique allows determining the absence of percolation in Zd : if p ≤ pc d, the critical point for independent bond percolation, percolation does not occur for all J, that is, p c d ≥ pc d (see [24], Proposition 3.6, for the proof of a more general statement). Actually, in the present case p c d = pc d (see Theorem 2.8 below). It is a different matter to establish if for a given (sufficiently large) p, percolation occurs for all J. A positive answer is given in Theorem 2.2 below, which is preceded by some definitions and a technical lemma. In order to discuss the simplest cases first, we now restrict to Z2 , in which case pc 2 = 1/2 (see [17] and [18]). The graph (Z2  BZ2 ) has a dual graph, having set of vertices  12  21  + Z2 and bonds between all pairs of vertices at distance 1. Each bond b of the dual graph crosses a bond of BZ2  which is called its image Ib and γ ⊂ BZ2  is 1786 E. DE SANTIS AND A. GANDOLFI the image of a self-avoiding dual circuit if I−1 γ is a (self-avoiding) circuit of the dual. Given a box  ⊂ BZ2 , the image γ ⊂  of a self-avoiding dual circuit contained in  separates  \ γ into two parts, an inner part, Intγ  say, and an outer part, Estγ , in the sense that each path from Intγ  to Estγ  must intersect γ; moreover, ∂ ⊂Estγ  ∪ γ. Next, fix J ∈  and a box . Let γ be the image of a dual circuit and let η\γ ∈ 0 1 \γ be a J-unfrustrated configuration in  \ γ. The next lemma states that given J and η\γ , there is at least one way of partitioning the bonds of γ into at most two subsets such that all the bonds in each subset can be simultaneously occupied without generating frustration. Given a subset A ⊂ γ, define ηA by   / γ  η\γ b if b ∈ if b ∈ A ηA b = 1   0 if b ∈ γ \ A A set of bonds A ⊂ γ is called conditionally frustrated if there exists a frustrated circuit π ⊂ ηA −1 1; A is called conditionally unfrustrated if there is no such circuit. Lemma 2.1. Given a box , the image of a self-avoiding dual circuit γ ⊂ , a configuration of interactions J and a configuration η\γ , J-unfrustrated in  \ γ, there exists at least one subset A of γ such that both A and γ \ A are conditionally unfrustrated. Proof. We recursively define subsets Ai ⊂ γ by examining, in an arbitrary fixed order, the bonds b1  b2      b γ of γ. Let A1 = b1 , which is obviously conditionally unfrustrated; then suppose that Ai−1 is conditionally unfrustrated and consider bi . If Ai−1 ∪ bi is conditionally unfrustrated then let Ai = Ai−1 ∪ bi ; otherwise let Ai = Ai−1 . By construction, A = A γ is conditionally unfrustrated, and the lemma is proved if B = γ \ A γ is also such. Suppose that it is not. Then there exists a frustrated circuit π in ηB −1 1; π can be made self-avoiding by successively removing unfrustrated s.a. loops (i.e., circuits) or by taking the first s.a. loop if one is found to be frustrated. Then π must necessarily satisfy π ∩ γ = k ≥ 2, with k even. In fact, it is not possible that π ⊂  \ γ, as η\γ is unfrustrated; therefore, π must contain bonds of γ; as π is self-avoiding, which is here an essential property, it must have nonempty intersection with both Intγ  and Estγ . Therefore, being a circuit, it must cross γ an even number of times. We can write π = bi1  πi1  i2  bi2  πi2  i3      πik−1  ik  bik  πik  i1 , where bij ∈ B, and πij  ij+1 ⊂  \ γ (here ik+1 = i1 ) are s.a. paths; we can take ik > ij , for j = 1     k − 1, in the above fixed order; moreover, k is even. After examining bik , each of the bij ’s has been examined and not added to Aij , so that Aij −1 ∪ bij is conditionally frustrated. Thus there exists a frustrated circuit BOND PERCOLATION IN FRUSTRATED SYSTEMS 1787 −1 bij  πj  ⊂ η−1 / πj and πj ⊂ η−1 A 1. Next, Ai −1 ∪bi 1 ⊆ ηA∪bi 1, where bij ∈ j j j form the circuit π̃ = π1  πi1  i2  π2  πi2  i3      πk−1  πik−1  ik  πk  πik  i1  which can always be formed by traversing each πj in a suitably chosen direction, that is, by replacing bij by πj in π. Note that π̃ ⊂ η−1 A 1 and let π̃˜ = bi1  π1  πi1  i2  bi2  bi2  π2  πi2  i3      πk−1  πik−1  ik  bik  bik  πk  πik  i1  bi1  then  b∈π̃ Jb = =  Jb b∈π̃˜    j=1k b∈bij  πj  Jb ×  b∈π Jb = −1k+1 = −1 which contradicts the construction of A. ✷ We now use one of the sets determined in Lemma 2.1 to show the occurrence of percolation for large enough p. Theorem 2.2. In Z2 , for all J ∈ −1 1 , if p > 8/9 then percolation occurs νJ p a.e., that is, pc 2 ≤ 8/9. Proof. For a set γ ∈ BZ2 , we indicate with γ ≡ 0 the event that ηb = 0 for all b ∈ γ. If with νJ p -probability 1 there are finitely many images of selfavoiding dual circuits γ such that γ ≡ 0, then percolation occurs, as discussed below. Given the image γ of a self-avoiding dual circuit and a box  ⊃ γ, Lemma 2.1 implies that for any configuration η\γ there exists a subset A of γ having the property that if none of the bonds of γ \ A is occupied, then any subset of the bonds of A can be occupied without creating frustration, and vice versa, exchanging the roles of A and γ \ A. Denoting A = m ∈ 0     γ , we have for each η\γ , νJ p  γ ≡ 0 η\γ  = (2.4) ≤ 1 − p γ ηγ  UJ η\γ ∨ηγ =1 m m k=0 k −1 1 pη γ −1 0 1 − pηγ 1 − p γ pk 1 − p γ −k + γ −m k=0 ≤ 1 − p γ γ −m + 1 − pm 0≤m≤ γ /2 1 − p ≤ 1 1 − p γ /2  2 sup γ −m k pk 1 − p γ −k 1788 E. DE SANTIS AND A. GANDOLFI It follows that also νJ p  γ ≡ 0 ≤ 12 1 − p γ /2 and taking  → ∞ we get νJp γ ≡ 0 ≤ 21 1 − p γ /2 for any weak limit νJ p . For a given k, the number of images of self-avoiding dual circuits with exactly k bonds containing the origin in their interior can easily be bounded by k3k . Then if p > 89 , 25 γ νJ p γ ≡ 0 ≤ k≥1 k3k 21 1 − pk/2 < ∞ by Borel–Cantelli, νJ p a.e. there are only finitely many images γ of selfavoiding dual circuits surrounding the origin such that γ ≡ 0. Taking, for instance, a vertex external to all such γ’s but in the boundary of at least one, then such vertex must be in an infinite cluster. ✷ The question arises whether it is possible to close the gap between 1/2 and 8/9 and determine (at least the existence of) a value p = p c = pc . In Section 3 a negative answer is suggested, and proved for several graphs, by giving explicit positive lower bounds on p c − pc . In greater generality, but with no explicit bounds, the inequality pc < pc is shown also in [5] and [7]. We now introduce the more interesting class of FK random cluster measures and discuss the extension of the above result to such measures and to general dimension d. To simplify the notation and unify the treatment of conditionally unfrustrated Bernoulli and FK random cluster measures, we introduce a parameter q = 1 2 (which in the spin interpretation describes the number of possible spin values). The conditionally unfrustrated Bernoulli measure is included in the definition which follows by taking the (somewhat trivial) case of q = 1, while q = 2 correponds to the FK measures. Other values of q are sometimes considered (such as integer q ≥ 3 in the Potts model), but, for simplicity of exposition, we do not discuss these cases. We also give definitions in Zd , but simple modifications allow making the same construction for other regular graphs (such as the planar triangular graph which is also briefly discussed in Section 4 of this paper). Let q = 1 2 be fixed. Let  = k be a box and let η ∈ H = 0 1  and ω ∈ , = 1 3 − 2q Vert ; then define 1 PJ  η  ω  = νJ P  η  and 2 PJ  η  ω  −1 = pη 1 −1 1 − pη ZJ  0 1η ∼ω  J  where 1η ∼ω  J equals 1 if we have ω xω yJ b = 1 for each bond b = q x y ∈ η−1  1, and equals 0 otherwise; and ZJ  is a normalizing factor. PJ  is a finite volume joint distribution of bond random variables η and of spin variables ω . We have not mentioned boundary conditions which are generally included in the definition (see, e.g., [24], pages 28–31), but our results are general enough to hold for all possible boundary conditions, and the required 1789 BOND PERCOLATION IN FRUSTRATED SYSTEMS modifications from our presentation, which only discusses the case with no boundary conditions, are immediate. q The interesting measures are the marginals of PJ  and their weak limits. −1 Given a configuration η ∈ H , the set η 1 defines, as before, connected components of the graph Vert  and thus clusters; we then let clη  be the number of clusters. We then have 26 q νJ p  η  = ω ∈, q PJ  η  ω  −1 = pη 1 −1 1 − pη 0 UJ η qclη  ZJ  and 27 µJ  β ω  = q η ∈H PJ  η  ω  = exp β i j∈ i−j =1 ZJ  Jij ωi ωj   where UJ η  equals 1 if η−1  1 is unfrustrated with respect to J on the graph Vert  , and equals 0 otherwise; β is such that p = 1 − e−2β , ZJ  is used to indicate the appropriate normalizing factor, possibly different from 1 one expression to the other and νJ p  = νJ p  is as in (2.1) (see, e.g., [24], Propositions 3.2 and 3.2, for a proof). For q = 1, the expression in (2.7) is trivial. For q = 2, the measure in (2.7) is the finite volume Gibbs random field for the interaction J, and the measure in (2.6) is the finite volume FK random cluster measure. We are then interested in the weak limits, as increasing subsequences of boxes  cover Zd in the sense of Van Hove, of the finite volq ume measures in (2.6) and (2.7), indicated hereafter by νJ p . In particular, for J ≡ 1, that is, Jb = 1 for all bond b, the measure (2.7) is the (finite volume) ferromagnetic Ising model with parameter β, where, in physical terms, β represents the inverse temperature. Equation (2.6) is, instead, the finite volume ferromagnetic FK random cluster model studied in [10]; in such cases, it is shown in [24] that the weak limit of (2.6) is unique. Let p2 d and pc2 d be defined as in (2.2) and (2.3) for the measures c 2 νJ p (from now on we indicate p1 d = p c d and pc1 d = pc d) and let c q q pc T and pc T be defined analogously for the triangular lattice. We remark that pc 2 d ≤ p2 c d as seen again by taking J s.t. J ≡ 1. In this case, the finite volume distributions are stochastically ordered (see [24]) and 2 2 thus νJ p (percolation occurs), where νJ p is the unique weak limit of the finite volume distributions (2.6), is monotone increasing in p. The next corollary extends the result of Theorem 2.2 to the FK random cluster measure in Z2 . Corollary 2.3. In Z2 , for all J ∈ −1 1 , if p > 16/17 then percolation occurs νJ p a.e., that is, pc2 2 ≤ 16/17. 1790 E. DE SANTIS AND A. GANDOLFI Proof. It is only required to modify (2.4), which now becomes 2 νJ p  γ ≡ 0 η\γ  ≤ 28 ≤ ≤ m m k=0 k sup 0≤m≤ γ /2 1 − p γ γ −m k=0 p/2k 1 − p γ −k + 1 − p γ −m 1  1 21 − p 2 2−p γ −m k p/2k 1 − p γ −k 1 − p γ − p/2m + 1 − pm 1 − p/2 γ −m γ /2  with γ ⊂ Z2 and η\γ as in (2.4), Therefore, 29 γ 2 νJ p γ̃ ≡ 0 ≤ 21 − p 2 2−p k1 k3 k  k/2  which is bounded if 21 − p/2 − p1/2 < 1/3, that is, p > 16/17. ✷ We now consider the extension of the above results to higher dimensions d. We have two results, one relevant for large d obtained by adapting [20], and the other useful for moderate dimensions, shown by comparison with dimension d = 2. We give some geometric notions that allow defining Peierls’ contours, which constitutes the generalization of dual circuits. We first consider the dual lattice of Zd , which can be taken as Zd + 1/2 1/2     1/2 A plaquette is then a unit d − 1-dimensional cube from the dual lattice centered at the middle of some bond (when considering the bond as a line connecting u to v embedded in Rd ) of the initial lattice Zd . Two plaquettes are called adjacent if they have a common d − 2-dimensional face, while, analogously, two bonds of the original lattice are called connected if they share a vertex. A set of plaquettes (or bonds) is called connected if any two of its elements belong to a chain of pairwise adjacent plaquettes (or bonds, respectively) from the set. As defined before, the boundary of a connected set 2 of bonds is the set of all bonds sharing a vertex with a bond not in 2. Let a connected set 2 of bonds be given. For every bond b in its boundary let us draw a plaquette orthogonal to b and intersecting it in the middle point: the set of such plaquettes form closed surfaces; each connected component γ is called Peierls’ contour and the number of its plaquettes is indicated by γ . Each Peierls’ contour γ separates thus Rd \ γ into a bounded and an unbounded part, and a Peierls’ contour containing the origin in the bounded part is called a Peierls’ contour including the origin. Note that in d = 2 Peierls’ contour corresponds exactly to images of s.a. dual circuits. By taking as γ a Peierls’ contour, Lemma 2.1 can be extended by repeating its proof verbatim. Lemma 2.4. Given a box , the image of a Peierls’ contour γ ⊂ , a configuration of interactions J and a configuration η\γ , J-unfrustrated in  \ γ, there 1791 BOND PERCOLATION IN FRUSTRATED SYSTEMS exists at least one subset A of γ such that both A and γ \ A are conditionally unfrustrated. It is shown in [20] that in Zd the number of different Peierls’ contours of size n including the origin is less than exp64nlog d/d. Therefore, Theorem 2.2 and Corollary 2.3 can be adapted to deal with dimensions d ≥ 3. The only change needed are in (2.5), which becomes γ νJ p γ ≡ 0 ≤ k≥1 exp64klog d/d 21 1 − pk/2 < ∞ and in (2.9), which becomes γ νJ p γ ≡ 0 ≤ exp64klog d/d k≥1  1 21 − p 2 2−p k/2 < ∞ By determing sufficient conditions for convergence of these series we get the following result. Lemma 2.5. In Zd , pc1 d ≤ 1 − exp−128log d/d and pc2 d ≤ 21 − exp−64log d/d  2 − exp−64log d/d The next result is once again stated in Zd for simplicity, but can easily be extended to any regular graph having a subgraph isomorphic to Z2 (see [19], page 10, for related definitions). Let Zd = x1      xd  xi ∈ Z and let ˜ be the graph having as vertices the vertices of Zd with xd = 0 and as bonds those bonds between them. Lemma 2.6. In Zd , pc1 d ≤ 8/9 and pc2 d ≤ 16/17. Proof. The idea is to apply the reasoning leading to (2.4), (2.5) or (2.8), (2.9), respectively, to the bonds in ˜ after conditioning to the configuration outside ˜. Let γ be a Peierls’ contour of Zd surrounding the origin O, and consider γ ∩ ˜, each connected component of which is the image of a s.a. dual circuit of Z2 . We denote by γ̃ the one such component surrounding O. Next, denote by Sγ the event that γ ≡ 0. If Sγ occurs for only finitely many γ’s then percolation occurs. Note that νJ p Sγ  = lim νJ p  Sγ  →∞ ˜ =  ∩ ˜; then νJ p  Sγ  = η ∈H νJ p  Sγ η\˜ νJ p  η\˜ . and let  ˜ ˜ \ \ Given η\˜ , if Sγ occurs for some Peierls’ contour γ then γ̃ is such that ˜ between Intγ̃  ˜ and Estγ̃ . ˜ (i) γ̃ ≡ 0, (ii) there are no connections in  \  1792 E. DE SANTIS AND A. GANDOLFI Note that if for all η\˜ the number of images γ̃ of s.a. dual circuits satisfying (i) and (ii) is bounded with νJ p -probability 1 (on 0 1 ˜), then percolation occurs. Given η\˜ , we denote the set of γ̃ such that (ii) occurs by Ŵ̃η\˜ and for a given γ̃ ∈ Ŵ̃η\˜ by Sγ̃ the event that also (i) occurs. To each γ̃ ∈ Ŵ̃η\˜ ˜ and Lemmas 2.1 and 2.4 apply since all possible connections between Intγ̃  ˜ Estγ̃  are through γ̃. Therefore, νJ p ¯ Sγ̃ η\˜  ≤ γ̃ Ŵ̃ νJ p ¯ γ̃ ≡ 0, ∈ η ˜ \ and, as in (2.5) or (2.9), for every η\˜ , νJ p Sγ̃ occurs for some γ̃ ∈ Ŵ̃η\˜ η\˜  ≤ k≥1 k3k 12 1 − p k /2 < ∞ or νJ p Sγ̃ occurs for some γ̃ ∈ Ŵ̃η\˜ η\˜  ≤ k3 k≥1 21 − p 2 2−p k1  k /2 < ∞ The bound on the right-hand side is independent of  and, taking  → ∞, one gets that percolation occurs with νJ p -probability 1 for p < 8/9 or 16/17, respectively. ✷ The next lemma shows that the occurrence of percolation implies occurrence of percolation from the origin. To simplify the statement we indicate by percolation almost everywhere a.e. the occurrence of percolation with νJ p probability 1 and by percolation from the origin the occurrence of percolation from the origin with positive νJ p -probability. Lemma 2.7. In the conditionally unfrustrated Bernoulli measure and in the FK random cluster measure on Zd percolation a.e. implies percolation from the origin. Proof. Suppose percolation occurs with νJ p -probability 1. Then, by σadditivity of the measure, there exists a vertex v ∈ V such that percolation occurs from v with positive νJ p -probability. Select a path π = b1  b2      bk  from v to the origin O and a box  ⊃ π. Fix a configuration η\π and successively examine the bonds of π recursively defining a set A. Consider b1 = v1 1  v1 2 ; if v1 1 and v1 2 are already connected in η−1 \π 1 then A1 = ⵰, otherwise let A1 = b1 . Define
η\π b if b ∈ / π η1 b = 1 if b ∈ A1  Suppose bi−1 has been analyzed and Ai−1 has been defined, and consider bi = vi 1  vi 2 . If vi 1 and vi 2 are already connected in ηi−1 −1 1 then let Ai = Ai−1 , otherwise let Ai = Ai−1 ∪ bi . Define
η\π b if b ∈ / π, ηi b = 1 if b ∈ Ai  BOND PERCOLATION IN FRUSTRATED SYSTEMS 1793 The set Aη\π  = A = Ak has the property that if all the bonds in A are occupied then v is connected to all vertices in π; moreover, all the bonds in A can be occupied simultaneously. Finally, since at each step only bonds between vertices not previously connected are added, the probability of such an occupation is at least p A . Let SO v be the event that O is connected to v, let Sv ∂ be the event that there is a connection from v to the boundary ∂ of  and let Sπ ∂ be the event that there is an occupied path included in  \ π from a vertex of π to ∂. Note that if Sv ∂ occurs, then either v is connected to ∂ by an occupied path included in  \ π, or a vertex of π is connected to ∂ by such path; this implies that if η ∈ Sv ∂ then η\π ∈ Sπ ∂ , so that νJ p Sπ ∂  ≥ νJ p Sv ∂  for every J and every p. Since also νJ p Sv ∂  > νJ p v percolates we have νJ p SO v ∩ Sv ∂  = ≥ η\π ⊂Sπ ∂ η\π ⊂Sπ ∂ νJ p SO v ∩ Sv ∂ η\π νJ p η\π  p Aη\π  νJ p η\π  ≥ p π νJ p Sv ∂  ≥ p π νJ p v percolates > 0 After taking the limit  → ∞ this shows that also percolation from the origin occurs with positive νJ p -probability. ✷ From now on, since we always establish the occurrence of percolation a.e., by “occurrence of percolation” we indicate both occurrence of percolation from the origin of the lattice with positive probability and occurrence of percolation with probability 1. Let pc d be the critical value of independent bond percolation in Zd , βc d be the critical point of the d-dimensional Ising model and βc T = 41 ln 3 ≈ 0276 be the critical point of the triangular lattice. Note that the exact √ value of βc d is known only in d = 2, in which case it is βc 2 = 21 ln1 + 2 (see √ [4], √ page 77), and thus we have percolation for no J if p < 1 − 1/1 + 2 = 2 − 2 ≈ 0586. The following theorem collects all the results proved in the previous lemmas. Only the statement (2.15) concerning the triangular lattice requires one additional modification, its proof being as the proof of Corollary 2.3 with k2k replacing k3k in (2.9). Theorem 2.8. Let d ≥ 2. We have 210 p1 d = pc d c and 211 pc1 d ≤ min 8 9   1 − exp−128log d/d  1794 E. DE SANTIS AND A. GANDOLFI For the FK random cluster measure we have p2 d = 1 − exp−2βc d c 212 and 213 pc2 d  16 21 − exp−64log d/d   ≤ min 17 2 − exp−64log d/d
Finally, for the FK random cluster measure on the triangular lattice T, defined as in (2.6) with T replacing Zd , we have 214 1 p2 T = 1 − exp−2βc T = 1 − √ ≈ 0423 c 3 and pc2 T ≤ 76  215 3. Percolation for almost all interactions and strict inequalities. In this section we assume that the interactions are independent and randomly chosen according to a Bernoulli distribution ℘ such that ℘ Jb = 1 = 21 = 1−℘ Jb = −1 for all bonds b. Densities of ferromagnetic bonds other than 1/2 can obviously be treated, but for simplicity we restrict the exposition to density 1/2, which corresponds to the standard spin glass model. Having selected and fixed one such J, we want to study, as in the last section, the occurrence of percolation of occupied bonds at different values of the occupation probability 1 p, with respect to a conditionally unfrustrated Bernoulli measure νJ p = νJ p 2 or to an FK random cluster measure νJ p . The interesting feature is that, in general, there are values of p such that percolation occurs for ℘ -almost all J’s but does not occur for some J’s. q To begin, we need to specify a map J → νJ p which, to all, or at least ℘ almost all, J’s assigns one of the measures in which we are interested. By following and adapting [2], [23], [24], [25] and [26], one could determine a q map which to ℘ -almost all J’s assigns a distribution on the νJ p ’s with the physically interesting property of being translationally covariant. However, such a construction would become somewhat cumbersome and obscure some of our results. Therefore, we follow a less general but much simpler construction close to the one in [13]. For given p and a box  consider the joint distribution of J and η in , q q Pp  J  η  = ℘ J νJ  p  η  q q Let Pp be any weak limit of Pp  along a suitable subsequence of boxes  conq q verging to Zd (or, with suitable modifications, to T). Next, let νJ p = Pp · J q be the conditional distribution of η given J and Pp edge be the marginal of q Pp on the η variables. We adopt here a slight abuse of notation, since, for q given J, the νJ p is not necessarily unique, nor does it necessarily coincide BOND PERCOLATION IN FRUSTRATED SYSTEMS 1795 with those defined in Section 2 (in the first place, the current definition is well given for ℘ -almost all J’s only). Nonetheless, the results of Section 2 hold for q all J’s and all weak limits νJ p however taken, and thus carry over to the possibly more restricted class of measures defined here. The main result of this section concerns the occurrence or absence of percolation ℘ -almost everywhere and is based on extending Theorem 3 in [13]. We obtain upper and lower bounds, which hold for ℘ -almost all J’s, for the conditional occupation probabilities. These bounds allow comparing our measures with the independent Bernoulli measure on the same graph, and then estimating the regions where percolation does or does not occur. The comparison between measures is based on the following inequality, already discussed in [14]: given two measures ν1 and ν2 on same graph  , if for all b ∈  1 2 1 2 and for all η\b  η\b ∈ 0 1  \b we have ν1 ηb = 1 η\b  ≥ ν2 ηb = 1 η\b  then ν1 stochastically dominates ν2 (see [11]) and ν1 percolation occurs ≥ ν2 percolation occurs. Lemma 3.1. For each bond b in Zd we have the following. For q = 1, p 1 31 inf Pp edge ηb = 1 η\b  ≥  η\b 2 1 sup Pp edge ηb = 1 η\b  ≤ p 32 η\b This implies that 33 if p > 2pc d percolation occurs 1 for the conditional distribution νJ p for ℘ -almost all J, 34 if p < pc d percolation does not occur 1 for the conditional distribution νJ p for ℘ -almost all J. For q = 2, 2 inf Pp edge ηb = 1 η\b  ≥ 35 η\b 2 sup Pp edge ηb = 1 η\b  ≤ 36 η\b p  2 p  2 − 2p + p2 This implies that 37 if p > 2pc d percolation occurs 2 for the conditional distribution νJ p for ℘ -almost all J, 38  if p < 1 + 2pc d − 1 + 4pc d − 4pc d2 /2pc d percolation does not occur for the conditional distribution 2 νJ p for ℘ -almost all J. 1796 E. DE SANTIS AND A. GANDOLFI The same results as in (3.5)–(3.8) hold for the triangular lattice with pc T replacing pc d. These values must be compared with those of Theorem 2.8. In particular, for q = 1, (3.4) does not say anything new with respect to (2.10); (3.3) is useless for d = 2 where pc d = 12 but suggests that percolation for almost all J occurs for the triangular lattice in a region potentially larger than that where percolation occurs for all J. For the FK random cluster model related to spin glasses, obtained for q = 2, notice the following. In d =√ 2, since pc 2 = 1/2, (3.8) holds, surprisingly, exactly the same bound of 2 − 2 as (2.2), although the two methods seem to be completely unrelated and (3.8) is only an upper bound. For other values of d it is not easy to compare (3.8) and (2.2). However, we see that in the triangular lattice, where (3.8) gives about 0520 and (2.14) about 0423, there does exist a new region. For p between those two values percolation occurs for ℘ -almost no J, but, as the value in (2.14) is exactly the one above which there is percolation for J̄ (with a J̄ ≡ 1), it does occur for some J. By improving upon (3.8), we see in Theorem 3.2 below that also in Z2 such a phenomenon occurs; on the other hand, we expect that in general there are also values of p for which percolation occurs for ℘ -almost all J but does not for some J. An example in which both phenomena occur is obtained, however, only for a particular distribution on J (no longer two-valued) in Theorem 4.2 below. Proof. Taking a sequence of boxes n along which the joint distribution converges we have, by the martingale convergence theorem, that q Pα p n 39 q Pα p edge ηb = 1 η\b  = lim n →∞ Jn ηb q limn Pα p n ηb ∨ ηn \b  Jn  Jn q limn Pα p n ηb ∨ ηn \b  Jn   therefore, upper and lower bounds on Jb 310 Jb q Pα p n ηb ∨ ηn \b  Jn  ηb q  Pα p n ηb ∨ ηn \b  Jn  which are independent from J\b and η\b are also bounds for the conditional q probabilities Pα p edge ηb = 1 η\b . Using next the inequality mentioned before the statement of the lemma, it is possible to show that the marginal distribution on the η’s is dominated or dominates a Bernoulli measure, so that, for appropriate choices of p, percolation occurs with probability zero or one. An application of Fubini’s theorem q yields that such a result holds also for the marginal νJ p for ℘ -almost all J (see [13] for further details). We now estimate the r.h.s. of (3.9) from below. Given b = x y , η\b and J\b , three situations can occur: either x and y are connected in η\b , and in such a case this connection is ferromagnetic or antiferromagnetic, or x and y 1797 BOND PERCOLATION IN FRUSTRATED SYSTEMS are not connected in η\b . It is easy to see that after reducing the common factors, the conditional probabilities in the three cases are bounded as follows (see [13] for details). For q = 1 2, q 311 Pp edge ηb = 1 η\b  J\b 
p/Za ≥ inf min  J p/Za + 1 − p1/Za + 1/Zf   p/Zf p/q   p/Zf + 1 − p1/Za + 1/Zf  p/q + 1 − p where Zf Za  = ZJ p  when J = J\b ∨ Jb = 1 J = J\b ∨ Jb = −1. The ratio Zf /Za remains to be estimated, but this can be done by taking inf and sup over η\b of the ratios of corresponding terms. Again, three situations can occur, giving rise to the following estimate: 
Z 1 1 ≤ a 1 − p = min 1 − p 1−p Zf 312
 1 1 ≤ max 1 − p 1 =  1−p 1−p therefore, 313 q Pp edge ηb = 1 η\b  J\b  ≥ min
p p/q  2 p/q + 1 − p  = p  2 To prove (3.2) and (3.6) we repeat the previous steps, reversing the inequalq ities. In particular, Pp edge is bounded by taking the sup in the r.h.s. of (3.11), with Za /Zf satisfying (3.12). Therefore, 1 314 Pp edge ηb = 1 η\b  J\b  
p  p/p + q1 − p  ≤ max p + 1 − p2 − p The corresponding statements for the triangular lattice are obtained by obvious substitutions. ✷ A small improvement upon (3.2), and upon (3.6), can be obtained by a better estimate of the ratio Za /Zf than that given in (3.12). Such an improvement is really relevant only in Z2 , where it shows the existence of p ∈ 0 1 such that percolation occurs for ℘ -almost no J but it does occur for some J. Although the improvement we obtain is very small, this result has some relevance since it proves that there is a strict inequality in the behavior of some quantities (here the probability of percolation) between the Ising model and spin glasses; altough widely expected, there were no rigorous proofs of any such strict inequality (recently, more general but less explicit inequalities have been discussed in [5] and [7]). 1798 E. DE SANTIS AND A. GANDOLFI Theorem 3.2. Consider an FK random cluster model in Z2 with distribution ℘ on  . If x ≈ 0588 equals the unique root in 0 1 of the equation 4 − 4x − 18x2 + 39x3 − 37x4 + 17x5 − 3x6 = 0 √ then for p ∈ 2 − 2 x percolation occurs for ℘ -almost no J but it does occur for some J. 315 √ Proof. As seen in Lemma 3.1, √ 2 − 2 is the critical point for percolation if J = J̄ ≡ 1, so that for p > 2 − 2 percolation occurs for J̄. To show that it does not occur for P-almost all J we follow the proof of (3.6). All steps are the same apart from the estimate of the ratio Za /Zf . To get a better estimate, we use a small set around the given bond b and take sums over η by separating the indices in this set. In details, let b = −1 0 0 0 be the given bond and let  = b 0 0 1 0  0 0 0 1  0 0 0 −1 . In the following formula we denote joint configurations such as JA ∨JB simply by JA  JB , and UJ η by Uη J for typographical reasons; furthermore, a single number in the indication of the configuration J indicates the value taken by Jb for the bond b under consideration. Given ′ ⊃  and J′ \b , if a configuration η′ \ is unfrustated in ′ \  with respect to the restriction J′ \ of J′ to ′ \  (i.e., if UJ′ \ η′ \  = 1) we write η′ \ ≈ J′ . Then we have Za /Zf = (3.16) ≥ η ′ −1 −1 pη′ 1 1 − pη′ 0 2clη′  Uη′  −1 J′ \b  η ′ −1 −1 pη′ 1 1 − pη′ 0 2clη′  Uη′  1 J′ \b  min η′ \ η′ \ ≈J′ η −1 −1 pη′ 1 1 − pη′ 0 2clη′  Uη′  −1 J′ \b  η −1 −1 pη′ 1 1 − pη′ 0 2clη′  Uη′  1 J′ \b   Given J′ \ and η′ \ , such that Uη′ \  J′ \  = 1, and given J\b , the configurations of H\b can be classified, regardless of Jb , as follows. Let H̄ = η\b ∈ H\b Uηb = 0 η\b  η′ \  J\b  J′ \  = 1   H1 = η\b ∈ H̄ both Uηb = 1 η\b  η′ \  Jb = 1 J\b  J′ \  = 1  and Uηb = 1 η\b  η′ \  Jb = −1 J\b  J′ \  = 1   317 Hf = η\b ∈ H̄ both Uηb = 1 η\b  η′ \  Jb = 1 J\b  J′ \  = 1  and Uηb = 1 η\b  η′ \  Jb = −1 J\b  J′ \  = 0   Ha = η\b ∈ H̄ both Uηb = 1 η\b  η′ \  Jb = 1 J\b  J′ \  = 0  and Uηb = 1 η\b  η′ \  Jb = −1 J\b  J′ \  = 1  Let also Hf0 Ha0  = η ∈ H  η\b ∈ Hf Ha  and ηb = 0 and Hf1 Ha1  = η1 ∈ H  η\b ∈ Hf Ha  and ηb = 1  1799 BOND PERCOLATION IN FRUSTRATED SYSTEMS Note that H̄ = H1 ∪ Hf ∪ Ha , and that only the η ’s such that η\b ∈ H̄ give a nonzero contribution to the sums in (3.16). If η\b ∈ H1 , then both terms η = ηb = 1 η\b  and η = ηb = 0 η\b  give a nonzero contribution to both numerator and denominator of the last expression in (3.16). We consider separately all of these terms except η \b , which is the configuration such that η\b ≡ 0; note that, after collecting a term which is kept fixed from now on, η \b  ηb = 1 contributes p1 − p3 /2 to the sums in (3.16) and η\b  ηb = 0 contributes 1 − p4 . Next, let ap = p1−p3 /2+1−p4 . If η\b  ηb = 1 ∈ Ha1 then it gives a nonzero contribution only to the numerator of the last term in (3.16), and we obtain a lower bound by removing it. We thus get from (3.16), indicating η = η\b ∨ ηb , and indicating 2 to the power of the number of different clusters of ′ intersecting  divided by 25 as 2clη  ≥ 1, Za /Zf (3.18) ≥ min min η′ \ ≈J′  ηb min η\b ∈H1 η\b =η \b η ∈Hf0 ∪Ha0 ηb −1 pη  p η−1  1 1 − p −1 0 clη′  η−1  0 2 2clη′  −1 pη 1 1 −1 − pη 0 2clη   1−p p + ap η ∈H1 f p + ap   1 1+  −1 So we have the following:    min min 1 η′ \ ≈J′  −1 1 − pη pη\b 1 1 − pη\b 0+1 2clηb =0η\b  + ap η ∈Hf0 ∪Ha0 ∪Hf1 ≥ 1 η−1 1 η−1 0 clη   1−p  2    −1    where the last inequality is obtained by disregading Ha0 and observing that the ratio between the sums in Hf0 and Hf1 is 1 − p/p. By observing that η ∈Hf1 319 −1 pη  1 −1 1 − pη 0 clη  2  ≤ 1 we get, from (3.1),  Za /Zf ≥ min 1 1 + =  1 − p + ap p −1 −1 2 − 2p + p1 − p3 2 − p = Lp ≤ 1 2 + p1 − p3 2 − p Exchanging the roles of Za and Zf , one can see that this is also a lower bound for Zf /Za . Therefore, from the r.h.s. of (3.11) and from (3.19) we get  p p 2 Pα p edge ηb = 1 η′ \b  J′ \b  ≤ sup  p + 1 − p1 + Lp p + 21 − p p =  p + 1 − p1 + Lp 1800 E. DE SANTIS AND A. GANDOLFI Percolation does not occur if p is such that p/p + 1 − p1 + Lp < 1/2, and since p/p + 1 − p1 + Lp is increasing, this is equivalent to saying that p is less than the root x of xx + 1 − xLx−1 = 1/2, that is, (3.15). ✷ 4. Other values of the interaction. In the previous sections the interaction J assumed only the values 1 and −1, but slightly different phenomena occur when J takes other values, as discussed in this section. First, we need to briefly redefine all the measures with which we are dealing to take into account the other values of J; the main difference is that now at p close to 1 the set S of bonds which has the tendency to be occupied is, among the unfrustrated sets, the one which maximizes b∈S Jb . As a consequence, we need some care in extending Theorem 2.2, which is based on occupying at least half of the bonds of a given circuit. In redefining the model, the value of p, which is related to the (conditional) probability that a bond b is open, must itself be influenced by the value of Jb ; frustration, on the other hand, still depends only on signJb . To allow the value Jb = 0, a reasonable convention is that in this case the bond b cannot be occupied; we include this in the definition of UJ . We consider Zd , as before, and let I⊂ R be the set of possible values of the interaction. Let  = IB , φJ p  1 = b∈ η b=1 1 − 1 − p Jb ,  φJ p  0 = b∈ η b=0 1 − p Jb and define for finite  ⊂ B and q = 1 2, 41 q νJ p  η  = φJ p  1 φJ p  0 UJ η qclη   ZJ p  where UJ η  is the indicator function that ηb = 0 if Jb = 0, η is not frustrated (in the sense used before) with respect to the configuration signJ ∈ −1 1 defined, if Jb = 0, by signJb = signJb , and ZJ p  is the partition function. q As before, νJ p  can be realized as marginal of a joint measure on η’s and ω’s which is a conditional Bernoulli measure, and we omit these details now. The definition (4.1) and in particular the form of the factors 1 − p Jb and 1 − 1 − p Jb derive from the interpretation of (4.1) as a way of defining spin glass models. In this case, in fact, pJb = 1 − exp−2β Jb , and if Jb = 1 we have p = p1 = 1 − e−2β ; the form used in (4.1) expresses pJb as a function of p = p1 . Of course, if I = −1 1 , we obtain the models used in the previous sections. Let ℘ , a distribution on  , be a product of independent identical distributions on I. Then one can repeat the construction of Section 3 starting from q (4.1), to obtain a joint distribution Pp on  × ℋ . Some of the methods of Section 2 apply if the distribution of J is bounded away from 0. As an example, we have the following. Lemma 4.1. For any fixed dimension d, suppose there exists t > 0 such that for every bond b we have Jb ≥ t. Then for all J, percolation of occupied bonds does occur if p > 1 − 1/8q + 11/t  in the conditionally unfrustrated 1801 BOND PERCOLATION IN FRUSTRATED SYSTEMS Bernoulli model and in the FK random cluster model on Zd , that is, pq c d ≤ 1 − 1/8q + 11/t . In the triangular lattice, pc T ≤ 1 − 1/71/t . Proof. Repeating the proof of Lemma 2.6 one gets, for q = 1 2,  1 − p Jb νJ p  γ̃ ≡ 0 η\γ̃  ≤ b∈γ̃ ×   ηγ̃  UJ η\γ̃ ∨ηγ̃ =1 b∈γ̃ ηγ̃ × (4.2) ≤ ≤ 1 − 1 − p Jb  q b=1   1 − p b∈γ̃ ηγ̃ b=0  1 + q − 11 − p Jb  q1 − p Jb b∈A   1 + q − 11 − p Jb  + q1 − p Jb b∈γ̃\A Jb −1  sup 0≤m≤ γ̃ /2 −1 1 1+q−11−pt m q1−pt  1 q1 − pt ≤ 2 1 + q − 11 − pt + 1+q−11−pt q1−pt γ̃ −m γ̃ /2  where the second inequality is realized for some particular A ⊆ γ̃ chosen as in (2.4), that is, according to Lemma 2.1; the last inequality follows from the monotonicity of x → 1 + q − 11 − px /q1 − px so that any value of Jb can be replaced by t. Then, if 3q1 − pt /1 + q − 11 − pt 1/2 < 1, which is 8q + 11 − pt < 1, that is, p > 1 − 1/8q + 11/t , we have  k/2 q1 − pt 1 < ∞ k3k νJ p  γ ≡ 0 ≤ 43 2 1 + q − 11 − pt γ̃ k≥1 Lemma 2.7 applies, since it is based on frustration, that is, on signJ only, and thus percolation both a.e. and from the origin occurs. The result for the triangular lattice is obtained by replacing 3k by 2k in (4.3). ✷ The next result shows that, for a particular graph and a particular choice of the values assumed by J, it is possible to identify four regions of the interval 0 1 such that if p is in these regions, percolation occurs for none, almost none, almost all or all of the J’s, respectively. We cannot determine whether there is a sharp transition between the second and the third region as, somewhat surprisingly, a proof that the occurrence of percolation is monotone in p is not available. In any case, we believe that the separation of the four regions, including sharp transition points, holds for a wide selection of graphs and of 1802 E. DE SANTIS AND A. GANDOLFI distributions of the J’s. We show the result for the FK random cluster model in the triangular lattice. Theorem 4.2. Let T be the triangular lattice, let δ ∈ 0 1 and t > 0 be real numbers and let J = Jb b∈T be i.i.d. random variables distributed according to ℘ with ℘ J = 1 = ℘ J = −1 = 1 − δ/2 and ℘ J = t = ℘ J = −t = δ/2. Given ε ∈ 0 1/7, it is possible to take δ and t small enough that for the FK random cluster model on T the following happens: √ if p < 1 − exp−2β 3 ≈ 0422 percolation T = 1 − 1/ c (4.4) occurs for no J,  √ if 1 − 1/ 3 < p < 1 + 2pc T − 1 + 4pc T − 4pc T2 / (4.5) 2pc T ≈ 0452, percolation occurs for some but ℘ -almost none of the J’s, (4.6) if 1 − 1/7 + ε < p < 1 − 1/31/2t percolation occurs for ℘ -almost all but not all of the J’s, (4.7) if 1 − 1/71/t < p percolation occurs for all J’s. Here βc T and pc T are the critical points, on T, of the Ising model and of independent percolation, respectively. Proof. Equation (4.4) is shown by comparing the finite volume FK random cluster measure νJ p  relative to each J with that of J̄, where J̄b = 1 for all b ∈ T. In fact, νJ p  is dominated by νJ̄ p  for all J (see (3.29) in [24]). By the exact solution of the Ising √ model on T, we get, as in (2.14), that if 3, then for no J percolation occurs, while if p < 1 − exp−2β T = 1 − 1/ c √ p > 1 − 1/ 3 then percolation occurs at least for J̄; this shows (4.4) and part of (4.5). To complete the proof of (4.5) we only need to show that, for p in the given interval, percolation occurs for almost no J. We proceed as in showing (3.2) and (3.6), just modifying the calculations in (3.14) and (3.12). 2 By some calculation, it is possible to give an upper bound to Pp edge ηb η\b  J\b  in the form of (3.14), in which the maximum is taken among the same terms as in (3.14) plus additional terms for the case Jb = t −t involving the new normalization functions Zt a and Zt f . For these it holds that inf 1 − pt  1 − p−t  1 ≤ Zt a /Zt f ≤ sup1 − pt  1 − p−t  1  Inserting these bounds in the above-mentioned upper bound, one can obtain 2 Pp edge ηb = 1 η\b  J\b 
1 − 1 − pt p  ≤ sup  2 − 2p + p2 1 − 1 − pt + 1 − pt 1 + 1 − pt   1 − 1 − pt  p/2 − p 1 + 1 − pt 1803 BOND PERCOLATION IN FRUSTRATED SYSTEMS all terms of which are, as t is taken small, bounded again by p/2 − 2p + p2  as when J takes only values ±1, so that the bound (3.6) remains the same. To show (4.6) we first find a configuration of J for which percolation occurs √ exactly if and only if p > 1 − 1/ 31/t : this is done by taking J̃ such that J̃b = t for all b ∈ T. For this configuration of the interactions, one can refer again to the exact solution of the Ising model on the triangular lattice, just with p√replaced by 1−1−pt ; the condition for percolation is thus 1−1−pt > 1 − 1/ 3, so that percolation does not occur for all J if p < 1 − 1/31/2t . The lower bound in (4.6) is obtained by showing that for δ small the presence of bonds with Jb = ±t does not much influence the estimate in (2.8) used to show (2.15). In fact, by an estimate from the theory of greedy lattice animals in [9], the maximal fraction of bonds with Jb = ±t in a self-avoiding path near the origin is asymptotically bounded by any power of δ = P J = t smaller than 1/2 with ℘ -probability 1, in the sense that for every c > 0 there exists a constant h > 0 such that 1 1Jb =±t ≤ hδ1/2+c max lim n→∞ n π  π is a sa chain π =n π⊆n b∈π ℘ -almost everywhere. The same bound applies to images of dual circuits surrounding the origin, since if they are of size n they must lie within n . Therefore, we can repeat the proof of (2.15) by modifying the estimate as follows. Let γ be the image of a dual circuit and let now γ̃ = b ∈ γ Jb = ±1 and γt = γ\ γ̃. 2 Then, we can obtain a bound for νJ p  γ ≡ 0 using the r.h.s. of (2.18). Given the estimate on the size of γ̃, we get, by eliminating from the sum all terms 2 1 γ 1−hδ1/2+c /2 . Let with η−1 γt 1 = 0, νJ p  γ ≡ 0 η\γ  ≤ 2 21 − p/2 − p γ indicate images of s.a. circuit surrounding the origin; then, if δ is small enough, 48 γ νJ p γ ≡ 0 ≤ k≥1 k2k−1 21 − p/2 − pk1−hδ 1/2+c /2 for ℘ -almost all J, since k2k−1 is an upper bound of the number of dual circuits in the triangular lattice; the r.h.s. of (4.8) is bounded if p > 1 − 1/7 + ε for ε small enough, so that percolation occurs as required. If ε and t are small √ enough, then 6/7 + ε < 1 − 1/ 31/t and (4.6) holds. Finally, (4.7) is proved in Lemma 4.1. ✷ The next theorem discusses the occurrence of percolation when J can take the value 0. Recall that bonds b in which Jb = 0 cannot be occupied, so that in this case we can only refer to percolation a.e., as the origin might be surrounded by bonds with Jb = 0. Theorem 4.3 considers an FK random cluster model on Zd (or on T). There exists p large enough that percolation occurs, if and only if the density of nonzero interactions is strictly above the percolation threshold pc d [or pc T for T]. Theorem 4.3. Consider a conditionally unfrustrated Bernoulli model or an FK random cluster model on Zd , d ≥ 2 (or on the triangular lattice T). There 1804 E. DE SANTIS AND A. GANDOLFI exists p < 1 such that percolation occurs for ℘ -almost all J if and only if ℘ Jb = 0 > pc d [or pc T, respectively]. Proof. We give the proof for Zd , which can be easily adapted to the triangular lattice. If p < 1 and PJb = 0 ≤ pc d, then the conditional probability of occupation of a bond b is strictly less than pc d, so that the joint distribution is dominated by a nonpercolating Bernoulli measure. An explicit estimate is given by selecting M > 0 such that P J < M J = 0 = 1/2 and computing, for q = 1 2, q Pp edge ηb = 1 η\b  q = Pp ηb = 1 Jb = 0 η\b    1 1 1 − 1 − pM ≤ + 2 2 1 − 1 − pM + q1 − pM PJb = 0 < pc d If ℘ Jb = 0 > pc d, we want to rescale the J variables: a schematic representation of the rescaled variables is shown in Figure 1. There exists Fig. 1. BOND PERCOLATION IN FRUSTRATED SYSTEMS 1805 δ > 0 such that ℘ Jb ∈ / −δ δ > pc d, and we focus on the bonds in which Jb ∈ / −δ δ for one such δ which is fixed from now on. For h ∈ N define the box Bh = −h hd ∩ Zd . Let k > h, with k h ∈ N, and for x ∈ Zd let τx = τ2kx denote, by a slight abuse of notation, the translation by 2k times the vector x; for b = x y ∈ BZd  define the rescaled variable >b ∈ 0 1 to be such that >b = 1 if and only if the following occurs. There is a path in J−1 R \ −δ δ ∩ τx Bk  ∪ τy Bk  connecting τx Bh  to τy Bh , and, furthermore, all paths connecting ∂τz Bh  to ∂τz Bk  are connected to each other within J−1 R \ −δ δ ∩ τz Bk , both for z = x and z = y. If >b = 1 we say that >b occurs. Following the results in [15], the construction in [1] and uniqueness of the infinite cluster in independent percolation, it is possible to find, for every ε > 0, suitable integers k and h such that for all b ∈ BZd  P>b = 1 > 1 − ε. From now on we focus on the renormalized bonds b ∈ Z2 , which correspond to a layer Z2 ×−k kd−2 in the original lattice; renormalized quantities are identified by decorating them with a bar. The second step is to observe that the renormalized block variables >b1 and >b2 are independent if b1 and b2 have no common end point, that is, they are one-dependent. By [22], for any given Bernoulli 0–1 variables on the edges BZ2  of Z2 there exists ε such that if P>b = 1 > ε for every b ∈ BZ2 , then the variables >b b∈BZ2  stochastically dominate the given Bernoulli variables. For Bernoulli variables with occupation probability p > 8/9, we have that the probability that there exists the image γ of a dual circuit surrounding the origin such that b γb = 0 > 1/2 is less than k−1 1−pk/2 , which is a bounded series. Therefore, it is possible to select k k3 k and h such that, with ℘ -probability 1, all large images of dual circuits γ surrounding the origin contain at least γ /8 bonds b such that >b = 1. Next choose an order of the bonds in each τx Bh , x ∈ Z2 . For a given renormalized bond b = x y ∈ Z2 , if >b occurs then there is a first bond b1 in τx Bh  which is connected to τy Bh , and a first b2 in τy Bh  connected to b1 in τx Bh . Among all s.a. paths in J−1 −δ δ connecting b1 and b2 one can select the one which comes lexicographically first. Note that if >b∗ occurs for some ∗ b = x∗  y∗ sharing an end point with b, say τx Bh  = τx∗ Bh , then the first bond in τx Bh  is the same: in fact, each bond which is reached by connections between τx Bh  and τy Bh  is also reached by connections between τx∗ Bh  and τy∗ Bh  as all s.a. paths from ∂τx∗ Bh  and ∂τx∗ Bk  are connected to each other. For a given configuration of interactions J let   RJ = π b  π b is the first s.a. path connecting b1 ∈ b >b occurs τx Bh  to b2 ∈ τ̄y Bh   It is on RJ that we find a percolation cluster of occupied bonds. In fact, let now ηZd \RJ be a configuration such that percolation of occupied bonds does not occur in Zd \ RJ . If for some ηRJ percolation does not occur in ηZd \RJ ∨ ηRJ 1806 E. DE SANTIS AND A. GANDOLFI then there exists the image of a dual hypersurface γ surrounding the box Bh such that γ ≡ 0. However, this implies that there exists the image of a dual self-avoiding circuit γ in Z2 , seen as reference space of the block variables, such that b is not active for all the b in γ for which >b occurs; we indicate this event by saying that γ is not active. Given J, ηZd \RJ and b = x y such that >b occurs, it is possible to classify b as follows: either all connections which can be realized between b1 and b2 are of a given sign (ferromagnetic or antiferromagnetic) or both signs can be realized; this last case is actually the easiest to deal with as it can satisfy both constraints. Given γ, ηZd \RJ and ηRJ \b∈γ π b , it is possible to apply Lemma 2.1 following the lines of Lemma 4.1. To estimate the probability that γ is not active, we can assume that on γ there are at least γ /2 bonds b for which >b occurs (as this does not happen only on a finite number of such γ and this is not relevant in our estimate). It is now possible to find a J-unfrustrated configuration ηRJ in which at least γ/4 of the bonds of γ are such that >b occurs and b1 and b2 are connected (without creating frustration). Applying the methods used in Lemma 4.1, one can obtain the estimate νJ p  γ is not active η\RJ  = ≤  ℛ b∈η−1 R 1 1 − 1 − p Jb b∈η−1 R 0 1 − p Jb q  b∈η−1 R 1 1 − 1 − p Jb b∈η−1 R 0 1 − p Jb q J  J  J  2k0 q1 − pδ 1 + q − 11 − pδ k0 J   γ /4 clη\RJ ηRJ   clη\RJ ηRJ    where k0 = 2k + 1d , RJ  = RJ ∩ ,  = ηRJ   UJ ηRJ   η\RJ  = 1 , and ℛ = ηRJ  ∈  γ is not active in ηRJ  ∨ η\RJ . Now take p large enough that 2k0 q1 − pδ /1 + q − 11 − pδ k0 1/4 < 1/3. Either with νJ p positive probability there is percolation in ηZd \RJ (and we are done) or else the convergence of the series analogous to that in (2.5) implies that, with νJ p probability 1, there are only finitely many separating dual hypersurfaces and percolation occurs. ✷ Finally, we discuss the case in which J is unbounded. The problem is then to show that when p is small enough percolation does not occur for P almost all J, in spite of the high density of occupied bonds. In fact, this is the case even if J can take the value ±∞ with a positive but not too high probability; to include this case into the discussion let us define bond occupation variables for the bonds b such that Jb = ±∞ only if P Jb = ∞ < pc d. In this case the definition is the natural one obtained when taking all ∞ values to be equivalent and is given as follows: the bonds b such that Jb = ∞ form finite clusters, and in each of these clusters C there is some subset S which is unfrustrated; let us occupy one such subset at random among all such subsets of maximal cardinality. BOND PERCOLATION IN FRUSTRATED SYSTEMS 1807 Theorem 4.4. Consider a conditionally unfrustrated Bernoulli model or an FK random cluster model on Zd , d ≥ 2 (or on the triangular lattice T). 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