CORRELATION INEQUALITIES OF GKS TYPE FOR THE POTTS MODEL 9 GEOFFREY R. GRIMMETT 0 0 2 Abstract. Correlation inequalities are presented for function- n als of a ferromagnetic Potts model with external field, using the a random-cluster representation. These results extend earlier in- J equalities of Ganikhodjaev–Razak and Schonmann, and yield also 2 GKS-type inequalities when the spin-space is taken as the set of 1 qth roots of unity. ] h p - h 1. Introduction t a m Our purpose in this brief note is to derive certain correlation in- [ equalities for a ferromagnetic Potts model. The main technique is the random-cluster representation of this model, and particularly the FKG 1 v inequality. Some, at least, of the arguments given here are probably 5 known to others. Our results generalize the work of Ganikhodjaev and 2 6 Razak, who have shown in [5] how to formulate and prove GKS in- 1 equalities for the Potts model with a general number q of local states. . 1 Furthermore, our Theorems 2.5 and 2.7 extend the correlation inequal- 0 ities of Schonmann to be found in [12]. 9 0 : v 2. The inequalities i X Let G = (V,E) be a finite graph, and let J = (J : e ∈ E) and e r a h = (hv : v ∈ V) be vectors of non-negative reals, and q ∈ {2,3,...}. We take as local state space for the q-state Potts model the set Q = {0,1,...,q−1}. The Potts measure on G with parameters J has state space Σ = QV and probability measure 1   π(σ) = exp J δ (σ)+ h δ (σ) , Z  X e e X v v  e=hx,yi∈E v∈V   Date: first posted 29 July 2007, revised 7 January 2009. 1991 Mathematics Subject Classification. 82B20, 60K35. Key words and phrases. Griffiths inequality, GKS inequality, Ising model, Potts model, random-cluster model, angular spins. 1 2 GEOFFREY R. GRIMMETT for σ = (σ : v ∈ V) ∈ Σ, where δ (σ) = δ and δ (σ) = δ v e σx,σy v σv,0 are Kronecker delta functions, and Z is the appropriate normalizing constant. We shall make use of the random-cluster representation in this note, and we refer the reader to [9] for a recent account and bibliography. Consider a random-cluster model on the graph G+ obtained by adding a ‘ghost’ vertex g, joined to each vertex v ∈ V by a new edge hg,vi. An edge e ∈ E has parameter p = 1 − e−Je, and an edge hg,vi has e parameter p = 1 − e−hv. With φ the corresponding random-cluster v measure, we obtain the spin configuration as follows. The cluster C g containing g has spin 0. To each open cluster of ω other than C , g we allocate a uniformly chosen spin from Q, such that every vertex in the cluster receives this spin, and the spins of different clusters are independent. The ensuing spin vector σ = σ(ω) has law π. See [9, Thm 1.3] for a proof of this standard fact, and for references to the original work of Fortuin and Kasteleyn. Let f : Q → C. For σ ∈ Σ, let (2.1) f(σ)R = f(σ ), R ⊆ V. Y v v∈R Thinking of σ as a random vector with law π, we write hf(σ)Ri for the mean value of f(σ)R. Let F be the set of all functions f : Q → C such q that, for all integers m,n ≥ 0: (2.2) E(f(X)m) is real and non-negative, (2.3) E(f(X)m+n) ≥ E(f(X)m)E(f(X)n), where X is a uniformly distributed random variable on Q. That is, f ∈ F if each S = f(x)m is real and non-negative, and qS ≥ q m x∈Q m+n P S S . For i ∈ Q, let Fi be the subset of F containing all f such that m n q q (2.4) f(i) = max{|f(x)| : x ∈ Q}. This condition entails that f(i) is real and non-negative. Theorem 2.5. Let f ∈ F0. For R ⊆ V, the mean hf(σ)Ri is real- q valuedandnon-decreasingin the vectorsJ and h, andsatisfieshf(σ)Ri ≥ 0. For R,S ⊆ V, we have that hf(σ)Rf(σ)Si ≥ hf(σ)Rihf(σ)Si. If there is no external field, in that h ≡ 0, it suffices for the above that f ∈ F . q Theorem 2.6. Let q ≥ 2. The following functions belong to F0. q (a) f(x) = 1(q −1)−x. 2 GKS INEQUALITIES OF FOR THE POTTS MODEL 3 (b) f(x) = e2πix/q, a qth root of unity. (c) f : Q → [0,∞), with f(x) ≤ f(0) for all x. Case (a) gives us the inequalities of Ganikhodjaev and Razak, [5]. When q = 2, these reduce to the GKS inequalities for the Ising model, see [7, 8, 11]. We do not now if the implications of case (b) were known previously, or if they are useful. Perhaps they are elementary examples of the results of [6]. In case (c) with f(x) = δ , we obtain the first x,0 correlation inequality of Schonmann, [12]. Theorem 2.7. Let q ≥ 2, f ∈ F0, and let f : Q → C satisfy (2.2). 0 q 1 If f and f have disjoint support in that f f ≡ 0 then, for R,S ⊆ V, 0 1 0 1 hf (σ)Rf (σ)Si ≤ hf (σ)Rihf (σ)Si. 0 1 0 1 If h ≡ 0, it is enough to assume f ∈ F . 0 q Two correlation inequalities were proved in [12], a ‘positive’ inequal- ity that is implied by Theorem 2.6(c), and a ‘negative’ inequality that is obtained as a special case of the last theorem, on setting f (x) = δ 0 x,0 and f (x) = δ . We note that Schonmann’s inequalities were them- 1 x,1 selves (partial) generalizations of correlation inequalities proved in [4]. Amongst the feasible extensions of the above theorems that come to mind, we mention the classical space–time models used to study the quantum Ising/Potts models, see [1, 2, 3, 10]. 3. Proof of Theorem 2.5 Weusethecouplingoftherandom-clusterandPottsmodeldescribed in Section 2. Let E+ be the edge-set of G+, Ω+ = {0,1}E+, and ω ∈ Ω+. Let A ,A ,A ,...,A be the vertex-sets of the open clusters g 1 2 k of ω, where A is that of the cluster C containing g. g g Let R ⊆ V, and let f ∈ F0. By (2.1), q k f(σ)R = f(0)|R∩Ag| f(X )|R∩Ar|, Y r r=1 where X is the random spin assigned to A . This has conditional r r expectation k g (ω) := E(f(σ)R | ω) = f(0)|R∩Ag| E(f(X)|R∩Ar| | ω). R Y r=1 By (2.2) and (2.4), g (ω) is real and non-negative, whence so is its R mean φ(g ) = hf(σ)Ri. R We show next that g is a non-decreasing function on the partially R ordered set Ω+. It suffices to consider the case when the configuration 4 GEOFFREY R. GRIMMETT ω′ is obtained from ω by adding an edge between two clusters of ω. In this case, by (2.3)–(2.4), g (ω′) ≥ g (ω). That hσRi = φ(g ) is non- R R R decreasing in J and h follows by the appropriate comparison inequality for the random-cluster measure φ, see [9, Thm 3.21]. Now, k E(f(σ)Rf(σ)S | ω) = f(0)|R∩Ag|+|S∩Ag| E f(X)|R∩Ar|+|S∩Ar| ω . Y (cid:0) (cid:12) (cid:1) r=1 (cid:12) By (2.3), E(f(σ)Rf(σ)S | ω) ≥ g (ω)g (ω). R S By the FKG property of φ, see [9, Thm 3.8], hf(σ)Rf(σ)Si = φ E(f(σ)Rf(σ)S | ω) ≥ hf(σ)Rihf(σ)S)i, (cid:0) (cid:1) as required. When h ≡ 0, the terms in f(0) do not appear in the above, and it therefore suffices that f ∈ F . q 4. Proof of Theorem 2.6 We shall use the following elementary fact: if T is a non-negative random variable, (4.1) E(Tm+n) ≥ E(Tm)E(Tn), m,n ≥ 0. This trivial inequality may be proved in several ways, of which one is the following. Let T , T be independent copies of T. Clearly, 1 2 (4.2) (Tm −Tm)(Tn −Tn) ≥ 0, 1 2 1 2 since either 0 ≤ T ≤ T or 0 ≤ T ≤ T . Inequality (4.1) follows by 1 2 2 1 multiplying out (4.2) and averaging. Case (a). Inequality (2.4) with i = 0 is a triviality. Since f(X) is real- valued, withthesamedistributionas−f(X), E(f(X)m) = 0whenmis odd, and is positive when m is even. When m+n is even, (2.3) follows from (4.1) with T = f(X)2, and both sides of (2.3) are 0 otherwise. Case (b). It is an easy calculation that E(f(X)m) = 1{q divides m}, where1{F}istheindicatorfunctionofthesetF,and(2.2)–(2.3)follow. Case (c). Inequality (2.3) follows by (4.1) with T = f(X). GKS INEQUALITIES OF FOR THE POTTS MODEL 5 5. Proof of Theorem 2.7 We may as well assume that f 6≡ 0, so that f (0) > 0 and f (0) = 0. 0 0 1 We use the notation of Section 3, and write k (5.1) F (ω) = f (0)|R∩Ag| E(f (X)|R∩Ar| | ω), 0 0 0 Y r=1 k (5.2) F (ω) = E(f (X)|S∩Ar| | ω). 1 1 Y r=1 By (2.2), F and F are real-valued and non-negative. Since f ∈ F0, 0 1 0 q F is increasing. 0 Since f f ≡ 0, 0 1 E f (σ)Rf (σ)S ω = 1 (ω)F (ω)F (ω), 0 1 Z 0 1 (cid:0) (cid:12) (cid:1) (cid:12) where 1 is the indicator function of the event Z = {S = R ∪ {g}}. Z Here, as usual, we write U ↔ V if there exists an open path from some vertex ofU tosome vertex ofV. Let T bethesubset ofV containing all vertices joined to S by open paths, and write ω for the configuration T ω restricted to T. Using conditional expectation, (5.3) hf (σ)Rf (σ)Si = φ 1 F F 0 1 Z 0 1 (cid:0) (cid:1) = φ 1 F φ(F | T, ω ) , Z 1 0 T (cid:0) (cid:1) where we have used the fact that 1 and F are functions of the pair T, Z 1 ω only. On the event Z, F is an increasing function of the configura- T 0 tion restricted to V \T. Furthermore, given T, the conditional measure on V \T is the corresponding random-cluster measure. It follows that φ(F | T, ω ) ≤ φ(F ) on Z, 0 T 0 by [9, Thm 3.21]. By (5.3), hf (σ)Rf (σ)Si ≤ φ 1 F φ(F ) 0 1 Z 1 0 (cid:0) (cid:1) ≤ φ(F )φ(F ) = hf (σ)Rihf (σ)Si, 0 1 0 1 and the theorem is proved. When h ≡ 0, A = ∅ in (5.1), and it suffices that f ∈ F . g 0 q Acknowledgements The author is grateful to Jakob Bj¨ornberg, Chuck Newman, and Aernout van Enter for their comments and suggestions. 6 GEOFFREY R. GRIMMETT References [1] M. Aizenman and B. 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