Short proof of the sharpness of the phase transition for the random-cluster model with q=2
aa r X i v : . [ m a t h . P R ] D ec Short proof of the sharpness of the phase transition for therandom-cluster model with q = 2 Yacine AounUniversit´e de Gen`eveDecember 8, 2020
Abstract: The purpose of this modest note is to provide a short proof of the sharpness of thephase transition for the Random-cluster model with q = 2 by extending the approach developed byDuminil-Copin and Tassion [3] for q = 1. This in particular implies the exponential decay of the twopoint-correlation function in the subcritical Ising model. Let us start by defining the nearest neighbor random-cluster measure on Z d . For a finite subgraphΛ = ( V, E ) of Z d , a percolation configuration ω = ( ω ) e ∈ E is an element of { , } E . A configuration ω can be seen as a subgraph of Λ with vertex-set V and edge-set given by {{ x, y } ∈ E : ω x,y = 1 } . If ω x,y = 1, we say that { x, y } is open. Let k ( ω ) be the number of connected components in ω and o ( ω )(respectively f ( ω )) the number of open (respectively closed) edges in ω .Fix p ∈ [0 , , q >
0. Let µ Λ ,p,q be a measure defined for any ω ∈ { , } E by µ Λ ,p,q ( ω ) = q k ( ω ) Z p o ( ω ) (1 − p ) f ( ω ) ,where Z is a normalizing constant introduced in such a way that µ Λ ,p,q is a probability measure. Themeasure µ Λ ,p,q is called the random-cluster measure on Λ with free boundary conditions. For q ≥ 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 ) ni =0 in S such that v = 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 .For A ⊂ Z d , we write x S ↔ A for the event that x is connected in S to a vertex in A . If S = Z d , wedrop 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 p c ∈ [0 ,
1] satisfying µ Z d ,p,q (0 ↔ ∞ ) = ( = 0 if p < p c ,> p > p c . A very nice idea introduced in [3] is to define a new critical parameter ˜ p c for which it is easier to provesharpness, which will in turn imply that p c = ˜ p c (see Theorem 1 below). For a finite subset S of Z d ,let ∆ S be the set of edges with exactly one endpoint in S and define ϕ p ( S ) := p X { x,y }∈ ∆ S µ S,p,q (0 ↔ x ) . (1)Then define the following critical parameter 1 p c := sup { p ∈ [0 ,
1] : ϕ p ( S ) < S ⊂ Z d containing 0 } .The main theorem of this note is the following one. Theorem 1.
1. For p > ˜ p c , µ Z d ,p, (0 ↔ ∞ ) ≥ p − ˜ p c p .2. For p < ˜ p c , there exists c = c ( p ) > such that for every x ∈ Z d µ Z d ,p, (0 ↔ x ) ≤ exp( − c | x | ) . (2) Corollary 1.
We have p c = ˜ p c . In particular, the phase transition for the random-cluster model with q = 2 is sharp. Corollary 2.
In the Ising model, the two-point correlation function decays exponentially fast withdistance.
Corollary 1 follows directly from Theorem 1. Corollary 2 follows from the Edward-Sokal coupling (see[5]).
We will write µ Λ ,p instead of µ Λ ,p, . Let us start by proving the second item of Theorem 1. Firstly,we will need the following lemma. Lemma 1 (Modified Simon’s inequality) . Let S be a finite set of Z d containing . For every z / ∈ S , µ Z d ,p (0 ↔ z ) ≤ p X { x,y }∈ ∆ S µ S,p (0 ↔ x ) µ Z d ,p ( y ↔ z ) . (3)A similar inequality was proved for the Ising model in [3]. Lemma 1 follows from the latter by theEdward-Sokal coupling by remarking that tanh( − log(1 − p )) ≤ p .Fix p < ˜ p c and S a finite set containing 0 such that ϕ p ( S ) <
1. Let Λ n be the box of size n around0 for the norm | · | . Fix Λ L such that S ⊂ Λ L . Then, using Lemma 1, we can write µ Z d ,p (0 ↔ z ) ≤ p X { x,y }∈ ∆ S µ S,p (0 ↔ x ) µ Z d ,p ( y ↔ z ) ≤ ϕ p ( S ) max y ∈ Λ L µ Z d ,p ( y ↔ z ) . (4)Note that | y − z | ≥ | z | − L . If | y − z | ≤ L , we bound µ Z d ,p ( y ↔ z ) by 1, otherwise we apply (4) to y and z instead of 0 and z . Iterating ⌊| z | /L ⌋ this strategy yields µ Z d ,p (0 ↔ z ) ≤ ϕ p ( S ) ⌊| z | /L ⌋ ,which proves the second item of Theorem 1.We now turn to the proof of the first item of Theorem 1. Let p > ˜ p c and ∂ Λ n be the boundary ofΛ n . We will prove the following differential inequality. Lemma 2.
Fix p > ˜ p c . Then ddp µ Λ n ,p (0 ↔ ∂ Λ n ) ≥ p (1 − µ Λ n ,p (0 ↔ ∂ Λ n )) . (5)Integrating this inequality between ˜ p c and p and taking n to infinity yields the first item of Theorem1. We will therefore focus on proving Lemma 2. Let E (Λ n ) be the set of edges whose endpoints arein Λ n . We will need the following result. 2 emma 3. Let A be an increasing event depending on edges of Λ n only. Then ddp µ Λ n ,p,q ( A ) = X e ∈ E (Λ n ) µ Λ n ,p,q ( A | ω e = 1) − µ Λ n ,p,q ( A | ω e = 0) . (6)The proof is a straightforward computation. Recall that an edge e is pivotal for a configuration ω and an event A if ω ( e ) / ∈ A and ω ( e ) ∈ A , where ω ( e ) (respectively ω ( e ) ) is the same configuration as ω except maybe for e where we close the edge e in ω ( e ) (respectively open the edge e in ω ( e ) ). We canuse Lemma 3 and the FKG inequality to see that ddp µ Λ n ,p ( A ) = X e ∈ E (Λ n ) µ Λ n ,p ( A | ω e = 1) − µ Λ n ,p ( A | ω e = 0) ≥ X e ∈ E (Λ n ) µ Λ n ,p ( ω ( e ) ∈ A ) − µ Λ n ,p ( ω ( e ) ∈ A )= X e ∈ E (Λ n ) µ Λ n ,p (e pivotal for A) . Set A := { ↔ ∂ Λ n } . Define the following random set γ := { z ∈ Λ n : z not connected to Λ cn } . By inclusion of events, we get X e ∈ E (Λ n ) µ Λ n ,p (e pivotal for 0 ↔ ∂ Λ n ) ≥ X e ∈ E (Λ n ) µ Λ n ,p (e pivotal for 0 ↔ ∂ Λ n , = ∂ Λ n )= X S ∈ S X e ∈ E (Λ n ) µ Λ n ,p (e pivotal for 0 ↔ ∂ Λ n , γ = S ) , where we decomposed with respect to all possibilities for γ in the last line. Remark that γ = S and e = xy is pivotal for 0 ↔ ∂ Λ n if and only if γ = S , 0 S ↔ x and y / ∈ S . Moreover, the event { S ↔ x } is mesurable with respect to the edges in S and the event { γ = S } is mesurable with respect to theedges that have at least one endpoint outside of S . Finally, all the edges in ∆ S are closed. Thus µ Λ n ,p (e pivotal for 0 ↔ ∂ Λ n , γ = S ) = µ Λ n ,p (0 S ↔ x, γ = S ) = µ S,p (0 ↔ x ) µ Λ n ,p ( γ = S ),where the last equality follows from the Markov property. Plugging this into the inequality abovegives X S ∈ S X e ∈ E (Λ n ) µ Λ n ,p (e pivotal for 0 ↔ ∂ Λ n , γ = S ) = 1 p X S ∈ S X xy ∈ ∆ S pµ S,p (0 ↔ x ) µ Λ n ,p ( γ = S )= 1 p X S ∈ S ϕ p ( S ) µ Λ n ,p ( γ = S ) ≥ p µ Λ n ,p (0 = ∂ Λ n ) , where we used that ϕ p ( S ) ≥ p > ˜ p c . Therefore, by combining all the inequalities, we get (5),which finishes the proof of Lemma 2. 3 Concluding remarks
1. It is natural to ask whether this approach can be further generalized for bigger values of q . Theproof of (5) does not use the fact that q = 2 and is valid for all q ≥
1, which implies ˜ p c ( q ) ≥ p c ( q ).However, it is easy to see that the susceptibility with free boundary conditions is always infinite at˜ p c , i.e. P x ∈ Z d µ Z d , ˜ p c ,q (0 ↔ x ) = ∞ .But the susceptibility is known to be finite at p c for q > Z (see [4, 2]) and is conjectured to befinite for q > Z d with d ≥ q large enough, this result is proved in [6]). This in turn impliesthat p c < ˜ p c and therefore Corollary 1 is not longer true in these cases.2. The argument presented here can be extended to any finite-range coupling constants ( J x,y ) x,y ∈ Z d ,see [3].3. For infinite-range coupling constants decaying sub-exponentially fast, the second item of Theorem1 doesn’t hold. However, as in [3], one can still prove that Lemma 2 holds and that the susceptibilitywith free boundary conditions is finite for every p < ˜ p c , which implies ˜ p c = p c . One can then use thesame reasoning as in [1] to deduce that µ Z d ,p, (0 ↔ x ) ≤ cJ ,x for every p < p c and for some positiveconstant c depending on p . Note that the use of the exponential decay of the volume of the connectedcomponent of 0 in [1] can be replaced by the existence of S such that ϕ p ( S ) < References [1] Y. Aoun. Sharp asymptotics of correlation functions in the subcritical long-range random-clusterand Potts models, 2020. arXiv:2007.00116.[2] H. Duminil-Copin, M. Gagnebin, M. Harel, I. Manolescu, and V. Tassion. Discontinuity of thephase transition for the planar random-cluster and Potts models with q >
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