aa r X i v : . [ m a t h . P R ] D ec HEIGHT FUNCTION DELOCALISATIONON CUBIC PLANAR GRAPHS
PIET LAMMERS
Abstract.
The interest is in models of integer-valued height functions on shift-invariantplanar graphs whose maximum degree is three. We prove delocalisation for modelsinduced by convex nearest-neighbour potentials, under the condition that each potentialfunction is an excited potential , that is, a convex symmetric potential function V with theproperty that its second derivative satisfies V ′′ (0) ≤ . Examples of such modelsinclude the discrete Gaussian and solid-on-solid models at inverse temperature β ≤ log 2 ,as well as the uniformly random K -Lipschitz function for fixed K ∈ N . In fact, βV is an excited potential for any convex symmetric potential function V whenever β issufficiently small. To arrive at the result, we directly study the geometric percolationproperties of sets of the form { ϕ ≥ a } and { ϕ ≤ a } , where ϕ is the random heightfunction and a a constant. Along the same lines, we derive delocalisation for modelsinduced by convex symmetric nearest-neighbour potentials which force the parity of theheight of neighbouring vertices to be distinct. This includes models of uniformly randomgraph homomorphisms on the honeycomb lattice and the truncated square tiling, as wellas on the same graphs with each edge replaced by N edges linked in series. The latterresembles cable-graph constructions which appear in the analysis of the discrete Gaussianfree field. Introduction
Height functions have an important role within statistical physics: they appear naturallyin the study of several lattice models, such as the dimer models, percolation models, theIsing model, and the six-vertex model, and height functions are also increasingly studiedin their own right. A natural class of height function models is the class generated by convex nearest-neighbour potentials , and we restrict our attention to this class, even thoughresults have been obtained for non-convex potentials as well as for non-pair-interactions.Sheffield was the first to study this class in its full generality in his seminal work
RandomSurfaces [18], where many fundamental properties are derived. There is a dichotomy for themacroscopic behaviour of each convex nearest-neighbour model: either the variance of theheight difference ϕ ( y ) − ϕ ( x ) remains bounded uniformly over the choice of the vertices x and y , the so-called localised or smooth phase, or the variance of ϕ ( y ) − ϕ ( x ) is unboundedas the distance from x to y grows large, the delocalised or rough phase. In dimensiontwo, either class in the dichotomy is nonempty, and there is not, at present, a generalstrategy to decide on the class that a model belongs to: existing results on localisationand delocalisation rely on ad hoc arguments. For a parametrised family of models, thisdichotomy often appears as a form of phase transition: the six-vertex model with a = b = 1 and c ≥ , for example, is localised for c > , and delocalised for c ≤ .In dimension two, it is furthermore conjectured that delocalised models have the Gauss-ian free field as their scaling limit. This extends the previously mentioned dichotomy; thescaling limit of a localised model is trivial (or, alternatively, a Gaussian free field of zerovariance). The conjecture implies that a delocalised model must delocalise logarithmically :that the variance of ϕ ( y ) − ϕ ( x ) grows logarithmically in the distance from x to y . Thisimplied statement has been confirmed for several models, and no examples to the contrary Mathematics Subject Classification.
Primary 82B20, 82B41; secondary 82B30.
Key words and phrases.
Delocalisation, random surfaces, height functions, statistical mechanics. have been found to date. Full convergence to the Gaussian free field has been confirmedonly for the dimer model, owing to its integrability [15, 16, 12].Let us give an overview of delocalisation results for integer-valued height function modelsother than the dimer model. Delocalisation was first proved for the discrete Gaussian modeland for the solid-on-solid model at high temperature, in the landmark article of Fröhlichand Spencer on the Kosterlitz-Thouless transition in the XY model [10]. The proof relieson a connection with the Coulomb gas. The six-vertex model with a = b = 1 and c ≥ has been shown to localise for c > [4, 14], and to delocalise for c ≤ . Delocalisationwas first proven at several distinct points of the phase diagram [3, 11, 9, 2, 14], and morerecently for the entire range [7]. Delocalisation of the Loop O (2) model, that is, the randomone-Lipschitz function on the vertices of the triangular lattice, was first established for theparameter x = 1 / √ in [5], and later for the parameter x = 1 , the uniform case, in [13].It is known that the delocalisation is logarithmic for all examples that were mentionedthus far. For the six-vertex model and the random Lipschitz function on the triangularlattice, the proof of this fact relies on Russo-Seymour-Welsh-type arguments. (See [6] fora seperate proof of logarithmic delocalisation of square ice.) Let us finally mention thatfor planar models, delocalisation has been established by Sheffield for measures at a slope ,whenever the slope is not in the dual lattice of the invariance lattice of the underlyingmodel [18]. The proof of this fact relies on cluster swapping , which is a versatile methodin the analysis of all convex nearest-neighbour models.It appears to be the case that the proofs of recent delocalisation results (for the six-vertex model, the random Lipschitz function on the triangular lattice, and the generalresult of Sheffield) appeal directly to the planar nature of the underlying graph in order todemonstrate that certain random sets of vertices cannot (simultaneously) percolate. Theserandom sets are often defined in terms of some transformation of the height function ϕ ,which is chosen to employ the specific duality properties of that specific setting. We takethe exact same route here, although the choice of random sets is perhaps the most naturalone for the setting of height functions: we shall directly study the percolative behaviourof the sets { ϕ ≥ a } and { ϕ ≤ a } for a constant. This approach is very general, andit is possible that it leads to similar results for other models when combined with morespecialised arguments. The work of Sheffield [18] plays an important role in the proof.2. Main results
We study models of height functions on planar graphs. In order to prove delocalisation,we require both the graph as well as the height function model to exhibit certain symmetries.Let us start with an appropriate definition for the set of graphs that are studied.
Definition 2.1 (shift-invariant planar graphs) . A shift-invariant planar graph is a pair ( G , L ) where L is a lattice in R and G = ( V , E ) a connected graph which has a locallyfinite planar embedding in R that is invariant under the action of L . The terms bipartite and maximum degree have their usual meaning; in the bipartite case, it is tacitly understoodthat L is chosen such that the two parts of the bipartition are invariant sets.Of particular interest are the shift-invariant planar graphs of maximum degree three.Prime examples of such graphs include the honeycomb lattice and the truncated squaretiling (sometimes also called the octagonal tiling or Mediterranean tiling ), see Figure 1.The pair ( G , L ) shall denote a shift-invariant planar graph throughout this article. Thefollowing definition introduces the standard terminology of height functions. Definition 2.2 (height functions) . A height function is an integer-valued function onthe vertex set V . Write Ω for the set of height functions, and write F for the Borel σ -algebra of the product topology on Ω = Z V . Identify each element θ ∈ L with the map V → V , x x + θ . For A ∈ F and θ ∈ L , let θA denote the event { ϕ ◦ θ : ϕ ∈ A } ∈ F . ELOCALISATION ON CUBIC PLANAR GRAPHS 3 (a) (b) (c)
Figure 1.
Examples of suitable graphs: (a) the honeycomb lattice, (b) thetruncated square tiling, and (c) the honeycomb lattice with edges replacedby series of edges.Write P (Ω , F ) for the set of probability measures on (Ω , F ) , and write P L (Ω , F ) for theset of measures which are furthermore shift-invariant , that is, measures µ which satisfy µ ◦ θ = µ for any θ ∈ L .To characterise the model of interest, we must introduce potential functions which en-code how a random height function ϕ behaves on a finite set Λ ⊂ V , conditional on thevalues of ϕ on the complement of Λ . We shall first introduce the standard notation inthe following definition, before describing exactly the class of potentials that are analysedhere. Definition 2.3 (potential functions, Hamiltonians, Gibbs measures) . Write λ for thecounting measure on Z . A potential function is a function V : Z → R ∪ {∞} with theproperty that the edge transition distribution given by e − V λ is a nontrivial finite measure.We shall also impose that V is symmetric, in the sense that V ( − x ) = V ( x ) for any x ∈ Z .For any finite set Λ ⊂ V , introduce the associated Hamiltonian defined by H Λ : Ω → R ∪ {∞} , ϕ X xy ∈ E (Λ) V ( ϕ ( y ) − ϕ ( x )) , where E (Λ) denotes the set of undirected edges in E that have at least one endpoint in Λ .A height function ϕ is called admissible if H Λ ( ϕ ) is finite for any Λ . If ϕ is admissible and Λ ⊂ V finite, then write γ Λ ( · , ϕ ) for the probability measure defined by γ Λ ( · , ϕ ) := 1 Z Λ ( ϕ ) e − H Λ ( δ ϕ | Vr Λ × λ Λ ) , (1)where Z Λ ( ϕ ) denotes a suitable normalisation constant. Thus, to sample from γ Λ ( · , ϕ ) , setfirst the random height function equal to ϕ on the complement of Λ , then sample its valueson Λ proportional to e − H Λ . The family ( γ Λ ) Λ with Λ ranging over the finite subsets of V is a specification. A measure µ ∈ P (Ω , F ) is called a Gibbs measure if it is supported onadmissible configurations, and if µ = µγ Λ for any finite Λ ⊂ V . Write G for the collectionof Gibbs measures, and G L for the collection of shift-invariant Gibbs measures, that is, G L := G ∩ P L (Ω , F ) .Let us now describe the class of potential functions that are studied. Definition 2.4 (excited potentials) . An excited potential is a convex symmetric potentialfunction V with the property that its second derivative satisfies V ′′ (0) ≤ . Note thatthe latter requirement is equivalent to asking that V ( ± ≤ V (0) + log 2 .The class of excited potentials is large, and includes the discrete Gaussian and solid-on-solid models at inverse temperature β ≤ log 2 , as well as the uniformly random K -Lipschitzfunction for fixed K ∈ N . In fact, if V is any convex symmetric potential function with PIET LAMMERS V ( ± < ∞ , then βV is an excited potential for β ≤ /V ′′ (0) . With the definition ofexcited potentials in place, we are ready to state the main result. Theorem 2.5 (delocalisation) . Let ( G , L ) be a shift-invariant planar graph of maximumdegree three, and let V denote an excited potential. Then the associated height functionmodel delocalises, in the sense that the set G L is empty. The fundamental feature of excited potentials is that they facilitate a form of symmetrybreaking on the edges on which ϕ is constant. In fact, the case for delocalisation is muchsimpler if the potential directly prohibits such edges from appearing. This motivates thefollowing definition. Definition 2.6 (parity potentials) . A parity potential is a symmetric potential function V which satisfies V ( x ) = ∞ for any even integer x , and whose restriction to the odd integersis convex.Indeed, the definition implies that any admissible height function ϕ has the propertythat ϕ ( x ) and ϕ ( y ) have a different parity for any edge xy ∈ E , which implies in particularthat the two values cannot be equal. The graph G must thus be bipartite when workingwith parity potentials, or no admissible height functions would exist. The prime exampleof a parity potential is the potential function defined by V ( x ) := ∞ · | x |6 =1 , which inducesa model of uniformly random graph homomorphisms. Theorem 2.7 (delocalisation) . Let ( G , L ) denote a bipartite shift-invariant planar graphof maximum degree three. Let V be a parity potential. Then the associated height functionmodel delocalises, in the sense that the set G L is empty. The theorem thus includes models of uniformly random graph homomorphisms on thehoneycomb lattice and the truncated square tiling. Another interesting construction isthe following: fix a natural number N , and replace each edge of the honeycomb latticeby N edges which are linked in series; see Figure 1. The previous theorem applies alsoto the model of uniformly random graph homomorphisms on this expanded graph. Thismodel is reminiscent of cable-graph constructions which appear in the analysis of thediscrete Gaussian free field, and the same construction works also for other graphs. Remarkthat Theorem 2.7 is included in [18] whenever there exists an automorphism of G thatinterchanges the two parts of the bipartition of G , see also [2].The proofs of Theorems 2.5 and 2.7 do not rely on the fact that the potential function V is the same for each edge; the theorems remain valid when replacing the Hamiltonianby H Λ : Ω → R ∪ {∞} , ϕ X xy ∈ E (Λ) V xy ( ϕ ( y ) − ϕ ( x )) , where xy V xy is a shift-invariant assignment of excited potentials or parity potentials tothe edges of the graph G . 3. Proof of the main results
The proof runs by contradiction. Throughout this section, G , L , and V are fixed, and—in order to arrive at a contradiction—we shall suppose that G L is nonempty, and fix someshift-invariant Gibbs measure µ ∈ G L . It is well-known that µ may be decomposed intoergodic components which are also shift-invariant Gibbs measures, and by doing so andchoosing one such component to replace µ , we may assume without loss of generality that µ is itself ergodic. This means that µ ( A ) ∈ { , } for any event A which satisfies θA = A for all θ ∈ L .We prove two delocalisation results: one for parity potentials, and one for excited po-tentials. The result for parity potentials follows from a simple geometrical argument. Theresult for excited potentials is harder to derive, and requires the introduction of external ELOCALISATION ON CUBIC PLANAR GRAPHS 5 randomness to break certain symmetries that appear. Subsection 3.1 describes this tech-nique for symmetry breaking in detail. Subsection 3.2 states a number of existing resultsfrom the work of Sheffield which play an important role in the argument. Subsection 3.3contains the general geometrical argument, which leads directly to the result for paritypotentials. Subsection 3.4 combines the ideas of Subsections 3.1 and 3.3, in order to derivedelocalisation for excited potentials.3.1.
Excited potentials and symmetry breaking.
Essential to the proof of Theo-rem 2.7 is the fact that the two endpoints of an edge cannot have the same height whenworking with a parity potential. This implies in particular that the height of at least one ofthe enpoints of an edge is nonzero. The same statement is simply false in the context of anexcited potential: it is perfectly possible that two neighbouring vertices have height zero.In this subsection we introduce a form of symmetry breaking which allows us to choose anonzero height for each such edge.Let V denote an excited potential, and suppose that V (0) = 0 without loss of generality(by adding a constant to the potential if necessary). Write V ∗ for the potential function V ∗ ( x ) := if x = 0 , log 2 if | x | = 1 , ∞ if | x | > .Then V ≤ V ∗ by definition of an excited potential. The relative weight exp − H Λ of eachconfiguration in the definition of γ Λ in (1) decomposes as a product of the factors e − V ( ϕ ( y ) − ϕ ( x )) over the edges xy ∈ E (Λ) . We shall further decompose the factor corresponding to a singlefixed edge xy ∈ E (Λ) , while keeping all other weights the same. First, decompose theweight corresponding to xy as the sum e − V ∗ ( ϕ ( y ) − ϕ ( x )) + h e − V ( ϕ ( y ) − ϕ ( x )) − e − V ∗ ( ϕ ( y ) − ϕ ( x )) i . (2)Note that either term is nonnegative. Introduce a new random variable, which indicatesif this edge is excited or not excited , with the weight corresponding to each state on theleft and right in (2). Formally, this requires the introduction of a probability kernel whichsamples the state of xy conditional on ϕ , but we refrain from doing so explicitly in order tokeep the discussion light. Let us make two simple observation. First, the state of xy andthe height function ϕ are independent conditional on ϕ ( x ) and ϕ ( y ) . Second, it almostsurely does not occur that the edge xy is not excited and simultaneously ϕ ( x ) = ϕ ( y ) . Inparticular, at least one of ϕ ( x ) and ϕ ( y ) is not equal to zero almost surely whenever theedge xy is not excited.Consider thus the measure γ Λ ( · , ψ ) , and condition the edge xy to be excited; the goal isto break symmetry for this excited edge. Focus on the weight on the left in (2). Note thatthe normalised edge transition distribution e − V ∗ λ/Z equals the distribution of the sum oftwo independent fair coin flips each valued ± / . If we write λ ∗ for the counting measureon the half-integers, then e − V ∗ ( ϕ ( y ) − ϕ ( x )) = 12 Z | ϕ ( x ) − z | = | ϕ ( y ) − z | = dλ ∗ ( z ) . Let V ∗ : Z + 1 / → R ∪ ∞ denote the convex symmetric potential function defined by V ∗ ( x ) := ∞ · | x |6 =1 / . We shall interpret the value of z as a new height associated with theedge e := xy . Thus, if we write ϕ ( e ) for z , then the previous equation becomes e − V ∗ ( ϕ ( y ) − ϕ ( x )) = 12 Z e − V ∗ ( ϕ ( e ) − ϕ ( x )) e − V ∗ ( ϕ ( e ) − ϕ ( y )) dλ ∗ ( ϕ ( e )) . PIET LAMMERS
This suggests that we may place a new vertex on the midpoint of e —labelled also e forconvenience—and replace the interaction V ∗ over the edge e with the interaction V ∗ overthe edges { x, e } and { y, e } . The marginal distribution of the heights on the original vertexset V is invariant under the introduction of this extra height ϕ ( e ) .If an edge is excited, then we may thus associate to it an extra height variable whichtakes values in the half-integers. In particular, this height is nonzero. We will exploit thisfeature by integrating this extra height variable into an exploration process that leads tothe delocalisation result. Note that conditional on ϕ ( x ) = ϕ ( y ) = 0 , the distribution of ϕ ( e ) is that of a fair coin flip with outcomes ± / , independent of all other randomness.This is the desired symmetry breaking.In the exploration process defined in Subsection 3.4, we will gradually reveal the values ofthe height function ϕ which is sampled from the measure γ Λ ( · , ψ ) . This exploration processrelies on external randomness to decide if edges are excited or not, and to decide on theedge heights whenever we choose to explore them. The measure γ Λ ( · , ψ ) —conditioned onthe exploration—changes as we reveal more heights and edge states. As long as factors ofthe form e − V ( ϕ ( y ) − ϕ ( x )) appear in the decomposition of this conditioned measure, we areable to apply the machinery introduced above: the ideas do not rely on the fact that thesample ϕ came from the unconditioned measure γ Λ ( · , ψ ) .3.2. Results from [18] . For fixed Λ ⊂ V , write F Λ for the σ -algebra generated by thefunctions ϕ ϕ ( x ) with x ranging over Λ . Write T for the intersection of F Λ over allcofinite subsets Λ of V . Events in T are called tail events . A measure is called extremal if it satisfies a zero-one law on T . We now quote a deep result from the work RandomSurfaces of Sheffield.
Theorem 3.1 (Sheffield, [18]) . Any measure µ ∈ G L which is ergodic is also extremal. We shall not employ this result on extremality directly. Instead, we shall quote twocorollaries, and explain broadly how they are derived. In the remainder of this subsection,we work in the context of a convex symmetric potential function V . The same corollarieshold true for parity potentials (which are convex over the odd integers ) after a technicalmodification which is explained in Subsection 3.3.Fix, throughout this section, some vertex r ∈ V , the root , and write Λ n for the set ofvertices in V which are at a graph distance at most n from r . By the topology of localconvergence we mean the coarsest topology on P (Ω , F ) that makes the map µ µ ( A ) continuous for any finite Λ ⊂ V and for any A ∈ F Λ . It follows from extremality of µ , that lim n →∞ γ Λ n ( · , ϕ ) = µ in the topology of local convergence for µ -almost every ϕ .It can be shown that for any finite set Λ ⊂ V and for any admissible height function ψ ,the measure γ Λ ( · , ψ ) satisfies the following two properties:(1) The distribution of ϕ ( r ) is log-concave,(2) The measure satisfies the FKG inequality (introduced formally below).These properties are preserved under taking limits in the topology of local convergence,which implies the following two results. Corollary 3.2 (Sheffield, [18]) . If µ is an extremal Gibbs measure, then the density of ϕ ( r ) is log-concave. Corollary 3.3 (Sheffield, [18]) . If µ is an extremal Gibbs measure, then it satisfies the FKG inequality , in the sense that µ ( f g ) ≥ µ ( f ) µ ( g ) for any pair of measurable functions f, g : Ω → [0 , which are increasing , meaning that f ( ϕ ) ≤ f ( ψ ) and g ( ϕ ) ≤ g ( ψ ) for anypair of height functions ( ϕ, ψ ) with ϕ ≤ ψ . ELOCALISATION ON CUBIC PLANAR GRAPHS 7
Log-concavity of the density of ϕ ( r ) implies in particular that ϕ ( r ) is integrable, whichis exactly the statement that we shall aim to contradict. The FKG property plays a rolein the construction of the contradiction.3.3. Proof of delocalisation for parity potentials.
In this subsection, V is a paritypotential, and G is required to be bipartite. Write { V , V } for the bipartition of the vertexset of G . We choose the labels such that ϕ takes even values on V and odd values on V almost surely—this is possible due to ergodicity of µ . Without loss of generality, wesuppose that r ∈ V . The two corollaries in the previous subsection remain true, exceptthat the density of ϕ ( r ) is log-concave over the even integers . In particular, it remainstrue that ϕ ( r ) is integrable. The contradiction—and therefore Theorem 2.7—shall followdirectly from integrability of ϕ ( r ) , together with the following lemma. The remainder ofthis subsection is dedicated to its proof. Lemma 3.4.
For any integer k , we have µ ( ϕ ( r )) ( k − , k + 1) . For any graph G and some random subset A of its vertices, write X G ( A ) for the eventthat the complement of A does not contain an infinite cluster. Consider, for example,critical site percolation on the triangular lattice, and write O and C for the set of openand closed vertices respectively. Then both X G ( O ) and X G ( C ) occur almost surely, eventhough neither O , nor C , percolates. In the setting of this article, we shall study the events X G ( { ϕ ≥ k + 1 } ) and X G ( { ϕ ≤ k − } ) . These events are shift-invariant, and thereforethey have probability zero or one for µ by ergodicity. They are related to the previouslemma through the following result. Note that the lemma holds true also when V is anyconvex symmetric potential, G any locally finite graph, and µ any Gibbs measure—as canbe seen from the proof. (The measure γ Λ ( · , ψ ) satisfies the FKG inequality in this moregeneral setting as well.) Lemma 3.5.
If the event X G ( { ϕ ≥ a } ) occurs almost surely, then µ ( ϕ ( r )) ≥ a , and if X G ( { ϕ ≤ a } ) occurs almost surely, then µ ( ϕ ( r )) ≤ a , for any a ∈ Z .Proof. We prove the first implication for a = 0 ; the other implications follow by symmetry(the parity of r and a does not play a role in the proof). The proof is straightforward, andrelies on the Markov property and on symmetry and convexity of the potential function V .For any finite Λ ⊂ V , write ∂ Λ for the vertices adjacent to Λ . Claim that the expectationof ϕ ( r ) is nonnegative in the measure γ Λ ( · , ψ ) , whenever Λ is a finite subset of V containing r , and ψ an admissible height function with ψ | ∂ Λ ≥ . Indeed, symmetry of the potentialfunction V implies that γ Λ ( ϕ ( r ) , ψ ) = − γ Λ ( ϕ ( r ) , − ψ ) , and the FKG inequality, the Markov property, and nonnegativity of ψ on ∂ Λ imply that γ Λ ( ϕ ( r ) , ψ ) ≥ γ Λ ( ϕ ( r ) , − ψ ) . This establishes the claim.Recall that Λ n denotes the set of vertices at a graph distance at most n from r . Explorethe values of ϕ in the following way. First reveal the values of ϕ on the complement of Λ n .Then, at each step, select a vertex that has not been revealed and which is adjacent to arevealed vertex on which the value of ϕ is negative. The exploration process terminateswhen no such vertex can be found. This occurs after at most finitely many steps, since Λ n is finite. Write R n = R n ( ϕ ) for the set of vertices that have not been revealed, and A n = A n ( ϕ ) for the event that r is contained in R n . Now µ (1 A n ϕ ( r )) = Z A n ( ψ ) γ R n ( ψ ) ( ϕ ( r ) , ψ ) dµ ( ψ ) ≥ PIET LAMMERS due to the previous claim—note that the values of ψ on ∂R n ( ψ ) are nonnegative. But µ ( A n ) → as n → ∞ since the event X G ( { ϕ ≥ } ) occurs almost surely. (cid:3) To prove Lemma 3.4, it now suffices to demonstrate that at least one of X G ( { ϕ ≥ k + 1 } ) and X G ( { ϕ ≤ k − } ) occurs with positive probability for the measure µ . Without lossof generality, we shall take k = 0 , and make a number of geometrical observations. (Notethat the parity of r no longer plays a role.) Recall that ϕ takes odd values on V almostsurely. Let G = ( V , E ) denote the unique graph which has V as its vertex set, andsuch that two vertices are neighbours if and only if they are at graph distance two in theoriginal graph G . The fact that G has maximum degree three implies directly that G isalso a shift-invariant planar graph. Moreover, as ϕ takes odd values on V , we observe thatthis set of vertices can be written as the disjoint union of { ϕ ≥ } ∩ V and { ϕ ≤ − } ∩ V .Write σ : V → {− , } for the unique map such that { σ = 1 } = { ϕ ≥ } ∩ V . The valueof σ ( x ) is called the spin of the vertex x .Since each Z -indexed path through G is also a path through G by restricting to halfits vertices, it is immediate that X G ( { σ = 1 } ) ⊂ X G ( { ϕ ≥ } ) and X G ( { σ = − } ) ⊂ X G ( { ϕ ≤ − } ) . Therefore it suffices to demonstrate that at least one of X G ( { σ = 1 } ) and X G ( { σ = − } ) has positive probability for µ . Consider now the converse of thisstatement. Then both { σ = 1 } and { σ = − } must percolate almost surely. Thus,to arrive at Lemma 3.4, it suffices to demonstrate that it is impossible that both setspercolate simultaneously with positive probability.For the contradiction we must first derive an intermediate result, namely that each of { σ = 1 } and { σ = − } contains one infinite cluster at most almost surely. This followsfrom the classical argument of Burton and Keane [1]. Lemma 3.6.
The cluster { σ = 1 } contains at most one infinite cluster almost surely.Proof. Since the law of ϕ is shift-invariant and since the distribution of ϕ ( x ) is log-concavefor any vertex x (over the odd or even integers, depending on the parity of x ), a unionbound implies that k ϕ | Λ n k ∞ /n goes to zero in probability as n → ∞ . In particular, lim inf n →∞ k ϕ | Λ n k ∞ /n = 0 almost surely.Recall that ∂ Λ denotes the set of vertices adjacent to Λ in the graph G . For fixed ϕ and n , write m n for the minimum of ϕ on ∂ Λ n , write ψ n for the function ψ n : V → Z ∪ {−∞} , x ( −∞ if x Λ n +1 , m n + d G ( x, ∂ Λ n ) if x ∈ Λ n +1 ,and write ϕ n for the height function ϕ ∨ ψ n . It is straightforward to see that | ϕ ( y ) − ϕ ( x ) | ≥ | ϕ n ( y ) − ϕ n ( x ) | ∈ Z + 1 for any edge xy ∈ E , which proves that ϕ n is admissible whenever ϕ is admissible. Moreover, ϕ n ≥ ϕ by construction. The previous paragraph implies furthermore that for fixed m ,the function ϕ n takes strictly positive values on Λ m for arbitrarily large values of n almostsurely. This construction is similar to a construction in [17, Section 14].Write N for the number of infinite clusters of { σ = 1 } . Remark that N is deterministicdue to ergodicity of µ . We aim to demonstrate that N ≤ almost surely. Consider firstthe case that ≤ N < ∞ . Then for m sufficiently large, the set Λ m intersects all infiniteclusters of { σ = 1 } with positive probability. But, conditional on this event, there existssome n such that ϕ n takes strictly positive values on Λ m almost surely. This implies that N = 1 with positive probability (by first sampling the height function, then resamplingits values on Λ n ), a contradiction. Finally, consider the case that N = ∞ almost surely.Then for m sufficiently large, Λ m intersects at least three infinite clusters of { σ = 1 } withpositive probability. Conditional on this event, ϕ n takes strictly positive values on Λ m for n sufficiently large almost surely. In particular, this implies that a trifurcation box in ELOCALISATION ON CUBIC PLANAR GRAPHS 9 the sense of the article of Burton and Keane [1] must occur for { σ = 1 } with positiveprobability, a contradiction. (cid:3) Let us collect the intermediate results obtained so far:(1) G = ( V , E ) is a shift-invariant planar graph,(2) σ is an ergodic distribution of spins which satisfies the FKG inequality,(3) { σ = 1 } and { σ = − } contain a single infinite cluster at most almost surely.In particular, the FKG inequality in the second statement follows from the fact that eachspin of σ is an increasing function of ϕ , combined with Corollary 3.3. It is known that theseintermediate results jointly rule out that both { σ = 1 } and { σ = − } contain an infinitecluster almost surely, due to Sheffield [18, Chapter 9]. We also refer to [8, Theorem 1.5]for an alternative proof. Thus, one of the two sets does almost surely not percolate, say { σ = − } , in which case the event X G ( { σ = 1 } ) occurs. This establishes the proof ofLemma 3.4 (through application of Lemma 3.5), and consequently that of delocalisationfor parity potentials (Theorem 2.7).3.4. Proof of delocalisation for excited potentials.
The letter V denotes an excitedpotential in this subsection and G is no longer required to be bipartite; all other notationsremain the same. Lemma 3.7.
For any integer k , we have µ ( ϕ ( r )) ( k, k + 1) .Proof. By the same reasoning as in the previous subsection, at least one of X G ( { ϕ ≤ k } ) and X G ( { ϕ ≥ k + 1 } ) must occur almost surely. Lemma 3.5 now yields the result. (cid:3) The remainder of this subsection is dedicated to the proof of the following lemma, whichobviously implies the desired contradiction.
Lemma 3.8.
For any integer k , we have µ ( ϕ ( r )) ( k − , k + ) . Set k = 0 without loss of generality. Let ϕ denote a sample from the Gibbs measure µ , and let c : E → {− / , / } denote an independent family of fair coin flips. Write σ : E → {− , } for the unique family of spins such that σ ( xy ) = 1 if and only if either max { ϕ ( x ) , ϕ ( y ) } > or ϕ ( x ) = ϕ ( y ) = 0 and c ( xy ) = 1 / . Each spin σ ( xy ) is an increasingfunction of ( ϕ, c ) , and therefore this family satisfies the FKG inequality. Note that by thesame arguments as in the proof of Lemma 3.6, the clusters { σ = 1 } and { σ = − } eachcontain a single infinite cluster at most almost surely. Since G is a shift-invariant planargraph of maximum degree three, we rule out the case that both { σ = 1 } and { σ = − } contain an infinite cluster almost surely by reasoning as in the previous subsection. In thefollowing argument we assume that the latter cluster does not percolate in order to provethat µ ( ϕ ( r )) ≥ / ; we later adapt the argument to the other case.Recall that Λ n denotes the set of vertices at a distance at most n from r . Sample a heightfunction ϕ from µ , and explore first the values of ϕ on the complement of Λ n where n issome large natural number. Conditional on these values, the law of ϕ is given by γ Λ n ( · , ϕ ) .Run the following exploration process. Select an edge e := xy which has not been selectedbefore, for which the height of x has been explored and ϕ ( x ) ≤ , and for which the heightof y has not been explored. Immediately explore the value of ϕ ( y ) whenever ϕ ( x ) < .First reveal the state of e whenever ϕ ( x ) = 0 , appealing to some external randomness in thecase that this event is not deterministic in the value of ϕ ( x ) and in the (unrevealed) valueof ϕ ( y ) . Immediately explore the value of ϕ ( y ) if e is not excited. Introduce the extra edgeheight ϕ ( e ) in the case that e is excited, and couple its value with the independent familyof coin flips in the following way. Note first that ϕ ( e ) = ϕ ( y ) / deterministically in thecase that the unrevealed height of ϕ ( y ) is not equal to zero. In the case that ϕ ( y ) = 0 , wecouple the height of ϕ ( e ) with c ( e ) such that they are almost surely equal. If ϕ ( e ) = − / then we explore the height of y , and otherwise we choose to not explore the value of y in this step. Repeat this process until no eligible edges are left. The process terminates afterfinitely many steps, since E (Λ n ) is finite.Write R n for the set of vertices in V that have not been revealed. Write E ∗ n for thecollection of directed edges ( x, y ) which satisfy x R n , y ∈ R n , and for which xy has beenrevealed to be excited. This implies automatically that ϕ ( xy ) = 1 / , by definition of theexploration. Write E n for the collection of undirected counterparts of edges in E ∗ n . Thenthe law of ϕ on V , conditional on the exploration, is given by Z e − H Rn,E ∗ n ( δ ϕ | Vr Rn × λ R n ) , (3)where H R n ,E ∗ n is the Hamiltonian H R n ,E ∗ n := X ( x,y ) ∈ E ∗ n V ∗ ( ϕ ( y ) − ) + X xy ∈ E ( R n ) r E n V ( ϕ ( y ) − ϕ ( x )) . Consider an edge xy ∈ E ( R n ) r E n . Then by definition of the exploration, either x and y are both in R n , or the height of the vertex which does not lie in R n is strictly positive.Thus, conditional on this exploration process, the behaviour of ϕ within R n is that of arandom surface with convex symmetric potential functions and fixed boundary conditionsof at least / . Recall from the proof of Lemma 3.5 that the Markov property, the FKGinequality, and symmetry of the convex potentials V ∗ and V implies that the expectationof ϕ ( x ) in the measure in (3) is at least / for each vertex x ∈ R n . Conclude that theexpectation of ϕ ( r ) is at least / , conditional on r ∈ R n .Since ϕ ( r ) is integrable for µ , it suffices to demonstrate that r ∈ R n with high probabilityas n → ∞ . If r R n , then some sequence of edges led to the exploration of r . In otherwords, there must exist a path ( x k ) ≤ k ≤ m through G which starts in ∂ Λ n and ends at aneighbour of r , such that ϕ ( x k ) ≤ for every vertex, and such that c ( x k x k +1 ) = − / forevery edge of this path for which ϕ ( x k ) = ϕ ( x k +1 ) = 0 . In particular, this implies that σ ( x k x k +1 ) = − for every edge of this path. Conclude that r ∈ R n whenever r is notconnected to ∂ Λ n for the subgraph { σ = − } ∪ E ( { r } ) of G . Since { σ = − } does notpercolate almost surely, the event r ∈ R n has high probability as n → ∞ . This concludesthe proof of Lemma 3.8 in the case that the random subgraph { σ = − } does not percolate.The argument for the case that the set { σ = − } percolates—and the set { σ = 1 } doesnot—is almost exactly the same. The only difficulty in the adaption of the proof stemsfrom the asymmetric definition of σ . This difficulty is avoided by the following observation.If ( x k ) ≤ k ≤ m is a path through G such that ϕ ( x k ) ≥ for each vertex and such that c ( x k x k +1 ) = 1 / for each edge that satisfies ϕ ( x k ) = ϕ ( x k +1 ) = 0 , then σ ( x k x k +1 ) = 1 foreach edge of that path. Since this is the only property of σ that is used in the second halfof the argument, we are done. Acknowledgements
The author thanks Hugo Duminil-Copin, Alex Karrila, Sébastien Ott, and Martin Tassyfor helpful comments and discussions. The author was supported by the ERC grantCriBLaM.
References
1. R. M. Burton and M. Keane,
Density and uniqueness in percolation , Comm. Math. Phys. (1989),no. 3, 501–505.2. Nishant Chandgotia, Ron Peled, Scott Sheffield, and Martin Tassy,
Delocalization of uniform graphhomomorphisms from Z to Z , arXiv preprint arXiv:1810.10124 (2018).3. Julien Dubédat, Exact bosonization of the Ising model , arXiv preprint arXiv:1112.4399 (2011).4. Hugo Duminil-Copin, Maxime Gagnebin, Matan Harel, Ioan Manolescu, and Vincent Tassion,
Dis-continuity of the phase transition for the planar random-cluster and Potts models with q > , arXivpreprint arXiv:1611.09877 (2016). ELOCALISATION ON CUBIC PLANAR GRAPHS 11
5. Hugo Duminil-Copin, Alexander Glazman, Ron Peled, and Yinon Spinka,
Macroscopic loops in theloop O ( n ) model at Nienhuis’ critical point , arXiv preprint arXiv:1707.09335 (2017).6. Hugo Duminil-Copin, Matan Harel, Benoit Laslier, Aran Raoufi, and Gourab Ray, Logarithmic vari-ance for the height function of square-ice , arXiv preprint arXiv:1911.00092 (2019).7. Hugo Duminil-Copin, Alex Karrila, Ioan Manolescu, and Mendes Oulamara,
Delocalization of theheight function of the six-vertex model , In preparation.8. Hugo Duminil-Copin, Aran Raoufi, and Vincent Tassion,
Sharp phase transition for the random-clusterand Potts models via decision trees , Ann. of Math. (2019), no. 1, 75–99.9. Hugo Duminil-Copin, Vladas Sidoravicius, and Vincent Tassion,
Continuity of the phase transitionfor planar random-cluster and Potts models with ≤ q ≤ , Comm. Math. Phys. (2017), no. 1,47–107.10. Jürg Fröhlich and Thomas Spencer, The Kosterlitz-Thouless transition in two-dimensional abelian spinsystems and the Coulomb gas , Comm. Math. Phys. (1981), no. 4, 527–602.11. Alessandro Giuliani, Vieri Mastropietro, and Fabio Lucio Toninelli, Haldane relation for interactingdimers , J. Stat. Mech.: Theory Exp. (2017), no. 3, 034002.12. ,
Height fluctuations in interacting dimers , Ann. Inst. H. Poincaré Probab. Statist. (2017),no. 1, 98–168.13. Alexander Glazman and Ioan Manolescu, Uniform Lipschitz functions on the triangular lattice havelogarithmic variations , arXiv preprint arXiv:1810.05592 (2018).14. Alexander Glazman and Ron Peled,
On the transition between the disordered and antiferroelectricphases of the 6-vertex model , arXiv preprint arXiv:1909.03436 (2019).15. Richard Kenyon,
Dominos and the Gaussian free field , Ann. Probab. (2001), no. 3, 1128–1137.16. , Height fluctuations in the honeycomb dimer model , Comm. Math. Phys. (2008), no. 3,675–709.17. Piet Lammers,
A generalisation of the honeycomb dimer model to higher dimensions , arXiv preprintarXiv:1905.13216 (2019).18. Scott Sheffield,
Random surfaces , Astérisque (2005).
Institut des Hautes Études Scientifiques
Email address ::