Sharp asymptotics of correlation functions in the subcritical long-range random-cluster and Potts models

For a family of random-cluster models with cluster weights q ≥ 1, we prove that the probability that 0 is connected to x is asymptotically equal to 1qχ(β) βJ0,x. The method developed in this article can be applied to any spin model for which there exists a random-cluster representation which is one-monotonic.

where Z is a normalizing constant introduced in such a way that µ Λ,β,q is a probability measure. The measure µ Λ,β,q is called the random-cluster measure on Λ with free boundary conditions. For q ≥ 1, the measures can be extented to Z d by taking the weak limit of measures defined in finite volume. We say that x and y are connected in S ⊆ Z d if there exists a finite sequence of vertices (v i ) n i=0 in S such that v 0 = x, v n = y and {v i , v i+1 } is open for every 0 ≤ i < n. We denote this event by x S ↔ y. If S = Z d , we drop it from the notation. We write 0 ↔ ∞ if for every n ∈ N, there exists x ∈ Z d such that 0 ↔ x and |x| ≥ n, where | · | denotes a norm on Z d .
For q ≥ 1, the model undergoes a phase transition: there exists β c ∈ [0, ∞] satisfying For β < β c , it follows from the definition that µ Z d ,β,q (0 ↔ x) goes to 0 as |x| goes to infinity. In [3], it was proved that if the coupling constants are finite-range, meaning that there exists R > 0 such that J x,y = 0 whenever |x − y| > R, then the probability of two points being connected decays exponentially fast in distance, i.e. for every β < β c , there exists c(β) > 0 such that for every x in Z d , µ Z d ,β,q (0 ↔ x) ≤ exp(−c|x|). (1.1) In this article, we consider the random-cluster models with infinite-range non-negative coupling constants (J x,y ) x,y∈Z d satisfying for every x, y, z ∈ Z d • H1 There exists c > 0 such that J 0,x ≤ cJ 0,y if |x| ≥ |y|.

Remark 1.
Important examples of such coupling constants are J 0,x = |x| −c with c > d and J 0,x = |x| − log |x| .
The main theorem of this article is the following one.
This theorem was already proved for q = 2 (the Ising model) in [11], and a weaker form of this theorem was proved for q = 1 (Bernoulli percolation) in [1]. They both relied on the Simon-Lieb inequality or its equivalent version for Bernoulli percolation (see [9] for the Ising model and [5] for the Bernoulli percolation). For q > 2, this inequality is not available, so those approaches cannot be extended. Instead of that, we are going to use the exponential decay of the size of the connected component of 0 that was recently proved in [8]. The latter used the so-called OSSS inequality introduced in [3]. This inequality was already used to prove sharpness in a lot of models (see [3,4,10]) for which there exists a random-cluster type representation which is one-monotonic. Therefore, we believe that the OSSS inequality coupled with the approach developed in this article has the potential to be applied to study subcritical phases of long-range spin models for which there exists a random-cluster representation which is one-monotonic (for instance the Ashkin-Teller model, see [12]).

Applications to the ferromagnetic q-state Potts model
The Potts model is one of the fundamental examples of a lattice spin model undergoing an ordered/disordered phase transition. It generalizes the Ising model by allowing spins to take one of q values, where q is an integer greater than or equal to 2.
The model on Z d is defined as follows. For a subset Λ of Z d , the probability measure is defined for any σ = (σ x ) x∈Λ ∈ {1, . . . , q} Λ by The model can be defined on Z d by taking the weak limit of measures in finite volume. The measure thus obtained is called the measure with free boundary conditions and is denoted by P Z d ,β,q . The Potts model undergoes a phase transition between the absence and the existence of long-range order at the so-called critical inverse temperature β c , see [7] for details. Our main theorem from the point of view of the Potts model is the following one.
Theorem 1.2. If (J x,y ) x,y∈Z d satifies H1-H5, then for q ≥ 1, β < β c and x ∈ Z d , Since the Potts model and the random-cluster models can be coupled (see [7]) in such a way that , Theorem 1.2 is a direct consequence of Theorem 1.1 and we will therefore focus on Theorem 1.1.

Backround
The following standard properties will be used in the proof of Theorem 1.1.
≤ βJ x,y . We refer to [2] for more details about this property.
Monotonicity of measures. The following is a standard consequence of the FKG inequality : for q ≥ 1, two subsets Λ 1 ⊂ Λ 2 of Z d and an increasing event A depending on the edges in Λ 1 (see [13] for definition of an increasing event and the proof of this inequality), we have Finally, the following non-trivial input will be a key ingredient of the proof.
This theorem was proved in [8].
Acknowledgments The author would like to warmly thank Hugo Duminil-Copin for his guidance and help through the master thesis as well as reading and pointing out mistakes in the previous versions of the present article. The author would also like to thank Yvan Velenik and Mallie Godard for helpful comments. Finally, the author would like to thank the Excellence Fellowship program at the University of Geneva for supporting him during his studies.
2 Proof of Theorem 1.1 . In the first case, this implies that the number of vertices in C(0) is big, which is unlikely to happen by (1.7). In order to make this idea precise, we introduce some notation. From now on, we will write µ instead of µ Z d ,β,q . Define f (x) := −2 log(J 0,x )/c 1 where c 1 is provided by Theorem 1.3. Denote by D y the event that the size of the connected component of y is smaller than f (x). Using the union bound, we can write

Upper bound
Using (1.5) we easily get that where we write o x (1) for a function that goes to 0 as |x| goes to infinity. This implies that the size of connected component of 0 can be assumed to be smaller than f (x). In this case, we are going to prove three lemmas. Lemma 2.1 and 2.2 give terms that are negligible with respect to J 0,x and Lemma 2.3 gives the sharp asymptotics. If 0 is connected x and the size of the connected component of x is smaller than f (x), then there must exist an open edge in C(0) whose endpoints are separated by a distance at least |x|/f (x). This will be an important observation in the proof of the next lemma. Before stating the lemma, we introduce some notation. For a configuration ω, define the random variable L(ω) := sup{|y 1 − y 2 | : y 1 , y 2 ∈ C(0), ω y1,y2 = 1} which gives the length of the biggest open edge in C(0). For y ∈ Z d and Λ ⊂ Z d , define R y (Λ) := sup{|x − y| : x ∈ Λ ∩ C(y)}. If Λ = C(y), we simply write R y . If L < ∞, then there exists an open edge {y 1 , y 2 } ∈ P 2 (Z d ) such that |y 1 − y 2 | = L. If there are several such edges, take the one that maximizes R y (C(y) \ {y 1 , y 2 }). If there are several such edges,then define an order ≺ on P 2 (Z d ) and choose the one minimal for ≺. In this case, we define R y := R y (C(y)), wherẽ C(y) := {x ∈ Z d : x is connected to y without using the edge {y 1 , y 2 }}.
Proof. Remark that if 0 is connected to x and L ≤ 1 2 |x|, then either R 0 ≥ 1 4 |x| or R x ≥ 1 4 |x|. Therefore, the union bound and the symmetry give The last equality follows from the fact that if 0 is connected to x, then C(0) = C(x). Let us make the following observation : if 0 is connected to x and |C(0)| ≤ f (x), then there exist y 1 , y 2 ∈ Z d satisfying This implies that L ≥ |x|/f (x). Therefore, there exists an open self-avoiding path (v i ) N i=0 from 0 to x such that there exists 0 ≤ l < N such that |v l − v l+1 | ≥ |x|/f (x). Take the smallest such l and set y 1 = v l and y 2 = v l+1 . For y, z ∈ Z d , we will write y ↔ 1 z if y is connected to z without using the edge {y 1 , y 2 }. Set k(x) := ⌊|x|/f (x)⌋. Finally, for n ∈ N, we set Λ n (y) := {x ∈ Z d : |x − y| < n}. Using the union bound, we get that Conditioning on {0 ↔ 1 y 1 } ∩ {y 2 ↔ 1 x} ∩ D 0 ∩ D x ∩ {R 0 ≥ 1 4 |x|} and using the finite energy property, we get with c given by H1. Plugging this into the inequality above gives In the last line, we used that |C(0)| ≤ f (x) on D 0 and |C(x)| ≤ f (x) on D x . Observe that if |C(0)| ≤ f (x) and R 0 ≥ 1 4 |x|, then there exists a, b ∈ Z d such that • 0 is connected to a in P 2 (Z d ) \ {a, b}, Set E a,b = P 2 (Z d ) \ {a, b}. Using the union bound, we get ←→ a, ω a,b = 1).
As before, the conditioning and the finite energy property give ←→ a).
Plugging this into the previous inequality gives In the second line, we used that (J x,y ) x,y∈Z d is invariant under translations. Therefore, combining all the inequalities we get Thanks to Lemma 2.1, we only need to estimate µ(0 ↔ x, C 0,x , L ≥ 1 2 |x|). Recall the definition of γ from H5. Using the union bound, we can write In the next lemma, we will focus on estimating the first term.
Lemma 2.2. For β < β c , 0 < γ < 1 given by H5 and x ∈ Z d Proof. If L ≥ 1 2 |x| and 0 is connected to x, then there exists an open self-avoiding path (v i ) N i=0 from 0 to x such that there exists 0 ≤ l < N such that |v l − v l+1 | ≥ 1 2 |x|. Set r(x) := ⌊ 1 2 |x|⌋. Reasoning as before, we get As in the proof of Lemma 2.1, we can prove can easily prove that Therefore, it follows from H1, H5 and Remark 2 that Lemma 2.2 implies by symmetry that We now focus on the next lemma, which gives the sharp asymptotics of the probability of 0 being connected to x. Lemma 2.3. For β < β c , 0 < γ < 1 given by H5 and The upper bound then follows by combining (2.1), (2.2), (2.3) and (2.4).
Proof. Set Λ = Λ |x| γ (0), Λ ′ = Λ |x| γ (x) and P 0,x := {R 0 ≤ |x| γ } ∩ {R x ≤ |x| γ }. Let Λ n be such that Λ, Λ ′ ⊂ Λ n . If 0 is connected to x, R 0 ≤ |x| γ and R x ≤ |x| γ , then there exists an open self-avoiding path (u i ) M i=0 from 0 to x such that there exists a unique 0 ≤ l < N such that |u l − u l+1 | ≥ |x| − 2|x| γ satisfying Set u l = y 1 , u l+1 = y 2 . Then, the union bound gives In the second inequality, we used the fact that on P 0,x , the number of connected components increases by 1 when ω y1,y2 goes from 1 to 0. In the third inequality, we used that 1 − exp(−βJ y1,y2 ) ≤ βJ y1,y2 . Fix ε > 0. It follows from H4 and the translational invariance that J y1,y2 ≤ (1 + ε)J 0,x since |y 2 − y 1 − x| ≤ δ|x| for |x| big enough. Therefore Now, if we decompose with respect to the possible connected components of x, the union bound gives: where the summation is over S containing x such that R x (S) ≤ |x| γ . The conditioning gives In the second inequality, we used the spatial Markov property (see [13,Chapter 3]) as well as the fact that if w ∈ S and z / ∈ S, then {w, z} is closed. In the third inequality, we used (1.4). Plugging this into the inequality above and taking the limit as n goes to infinity, we get for |x| big enough. We used the translational invariance in the second inequality. This finishes the proof of Lemma 2.3.
Therefore, the union bound gives The second inequality follows from the finite energy property and the third inequality from H4. Since δ ≤ 1/2 and