Exponential decay of loop lengths in the loop O(n) model with large n
Hugo Duminil-Copin, Ron Peled, Wojciech Samotij, Yinon Spinka
EEXPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODELWITH LARGE n HUGO DUMINIL-COPIN, RON PELED, WOJCIECH SAMOTIJ, AND YINON SPINKA
Abstract.
The loop O ( n ) model is a model for a random collection of non-intersecting loops onthe hexagonal lattice, which is believed to be in the same universality class as the spin O ( n ) model.It has been conjectured that both the spin and the loop O ( n ) models exhibit exponential decay ofcorrelations when n >
2. We verify this for the loop O ( n ) model with large parameter n , showingthat long loops are exponentially unlikely to occur, uniformly in the edge weight x . Our proofprovides further detail on the structure of typical configurations in this regime. Putting appropriateboundary conditions, when nx is sufficiently small, the model is in a dilute, disordered phase inwhich each vertex is unlikely to be surrounded by any loops, whereas when nx is sufficiently large,the model is in a dense, ordered phase which is a small perturbation of one of the three groundstates. Introduction
After the introduction of the Ising model [26] and Ising’s conjecture that it does not undergo aphase transition, physicists tried to find natural generalizations of the model with richer behavior.In [17], Heller and Kramers described the classical version of the celebrated quantum Heisenbergmodel where spins are vectors in the (two-dimensional) unit sphere in dimension three. Later,Stanley introduced the spin O ( n ) model by allowing spins to take values in higher-dimensionalspheres [34]. We refer the interested reader to [10] for a history of the subject.Formally, a configuration of the spin O ( n ) model on a finite graph G is an assignment σ ∈ Ω :=( √ n · S n − ) V ( G ) of spins to each vertex of G , where S n − ⊆ R n is the ( n − √ n serves as a convenient normalization. The Hamiltonian of the modelis defined by H G,n ( σ ) := − (cid:88) { u,v }∈ E ( G ) (cid:104) σ u , σ v (cid:105) , where (cid:104)· , ·(cid:105) denotes the scalar product in R n . At inverse temperature β , we define the finite-volumeGibbs measure µ G,n,β to be the probability measure on Ω given by dµ G,n,β ( σ ) := 1 Z spin G,n,β exp [ − β H G,n ( σ )] dσ, where Z spin G,n,β , the partition function , is given by Z spin G,n,β := (cid:90) Ω exp [ − β H G,n ( σ )] dσ (1)and dσ is the uniform probability measure on Ω (i.e., the product measure of the uniform distribu-tions on √ n · S n − for each vertex in G ).By taking the weak limit of measures on larger and larger subgraphs of an infinite planar lattice,such as Z or the hexagonal lattice H , an infinite-volume measure µ n,β can be defined, and one may Date : July 23, 2018.Research of H.D.-C. is supported by Swiss FNS, the ERC AG COMPASP and the NCCR SwissMap.Research of R.P. and Y.S. is partially supported by an ISF grant and an IRG grant.Research of W.S. is partially supported by an ISF grant. a r X i v : . [ m a t h - ph ] O c t HUGO DUMINIL-COPIN, RON PELED, WOJCIECH SAMOTIJ, AND YINON SPINKA ask whether a phase transition occurs at some critical inverse temperature. From this point of view,the behavior of the model is very different for different values of n : • For n = 1, the model is simply the Ising model, which is known to undergo a phase transitionbetween an ordered and a disordered phase, as proved by Peierls [32] (refuting Ising’s conjec-ture). The critical inverse temperature has been computed for the square and the hexagonallattices and it is fair to say that a lot is known about the behavior of the model. We referthe reader to [11, 13, 31] and references therein for an overview of the recent progress on thesubject. • For n = 2, the model is the so-called XY model (first introduced in [36]). Since the spinspace S is a continuous group, the Mermin–Wagner theorem [28] guarantees that there isno phase transition between ordered and disordered phases. Still, a Kosterlitz–Thoulessphase transition occurs as proved in [15, 22, 27, 35]. That is, below some critical inversetemperature, the spin-spin correlations µ n,β [ (cid:104) σ u , σ v (cid:105) ] decay exponentially fast in the distancebetween u and v , while above this critical inverse temperature, they decay only like an inversepower of the distance. • For n ≥
3, it is predicted that no phase transition occurs [33] and that spin-spin correlationsdecay exponentially fast at every positive temperature. The n = 3 case, corresponding tothe classical Heisenberg model, is of special interest. Let us mention that this predictionis part of a more general conjecture asserting that planar spin systems with non-Abeliancontinuous spin space do not exhibit a phase transition. As of today, the n ≥ /n expansion is performed as n tends to infinity.On the hexagonal lattice H , the spin O ( n ) model can be related to the so-called loop O ( n ) model introduced in [9]. Before providing additional details on the relation, let us define the loop O ( n ) model. A loop is a finite subgraph of H which is isomorphic to a simple cycle. A loopconfiguration is a spanning subgraph of H in which every vertex has even degree; see Figure . The non-trivial finite connected components of a loop configuration are necessarily loops, however, a loopconfiguration may also contain isolated vertices and infinite simple paths. We shall often identifya loop configuration with its set of edges, disregarding isolated vertices. In this work, a domain H is a non-empty finite connected induced subgraph of H whose complement V ( H ) \ V ( H ) inducesa connected subgraph of H (in other words, it does not have “holes”). For convenience, all of ourresults will be stated for domains, although the definitions and techniques may sometimes be appliedin greater generality. Given a domain H and a loop configuration ξ , we denote by LoopConf ( H, ξ )the collection of all loop configurations ω that agree with ξ on E ( H ) \ E ( H ). Finally, for a domain H and a loop configuration ω , we denote by L H ( ω ) the number of loops in ω which intersect E ( H )and by o H ( ω ) the number of edges of ω ∩ E ( H ). Definition 1.1.
Let H be a domain and let ξ be a loop configuration. Let n and x be positivereal numbers. The loop O ( n ) measure on H with edge weight x and boundary conditions ξ is theprobability measure P ξH,n,x on LoopConf ( H, ξ ) defined by P ξH,n,x ( ω ) := x o H ( ω ) n L H ( ω ) Z ξH,n,x , ω ∈ LoopConf ( H, ξ ) , where Z ξH,n,x is the unique constant which makes P ξH,n,x a probability measure. We note that the loop O ( n ) model is defined for any real n > O ( n ) model isonly defined for positive integer n (the loop O ( n ) model may be defined also with n = 0 by takingthe limit n →
0, giving rise to a self-avoiding walk model). Let us now briefly discuss the connectionbetween the loop and the spin O ( n ) models (with integer n ) on a domain H ⊂ H . Rewriting the XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n partition function Z spin H,n,β given by (1) using the approximation e t ≈ t gives Z spin H,n,β = (cid:90) Ω (cid:89) { u,v }∈ E ( H ) e β (cid:104) σ u ,σ v (cid:105) dσ ≈ (cid:90) Ω (cid:89) { u,v }∈ E ( H ) (1 + β (cid:104) σ u , σ v (cid:105) ) dσ = (cid:88) ω ⊂ E ( H ) β o H ( ω ) (cid:90) Ω (cid:89) { u,v }∈ E ( ω ) (cid:104) σ u , σ v (cid:105) dσ. The integral on the right-hand side equals n L H ( ω ) if ω ∈ LoopConf ( H, ∅ ) and 0 otherwise; seeAppendix A for the calculation. Here, the normalization of taking spins on the sphere of radius √ n is used. Hence, substituting x for β , Z spin H,n,x ≈ (cid:88) ω ∈ LoopConf ( H, ∅ ) x o H ( ω ) n L H ( ω ) = Z ∅ H,n,x . In the same manner, the spin-spin correlation of u, v ∈ V ( H ) may be approximated as follows. µ H,n,x [ (cid:104) σ u , σ v (cid:105) ] = (cid:90) Ω (cid:104) σ u , σ v (cid:105) exp [ − x H H,n ( σ )] dσZ spin H,n,x ≈ n · (cid:88) λ ∈ LoopConf ( H, ∅ ,u,v ) x o H ( λ ) n L (cid:48) H ( λ ) J ( λ ) (cid:88) ω ∈ LoopConf ( H, ∅ ) x o H ( ω ) n L H ( ω ) , (2)where LoopConf ( H, ∅ , u, v ) is the set of spanning subgraphs of H in which the degrees of u and v are odd and the degrees of all other vertices are even. Here, for λ ∈ LoopConf ( H, ∅ , u, v ), o H ( λ ) isthe number of edges of λ , L (cid:48) H ( λ ) is the number of loops in λ after removing an arbitrary simplepath in λ between u and v , and J ( λ ) := nn +2 if there are three disjoint paths in λ between u and v and J ( λ ) := 1 otherwise (in which case, there is a unique simple path in λ between u and v ); seeAppendix A for the calculation.Unfortunately, the above approximation is not justified for any x >
0. Nevertheless, (2) provides aheuristic connection between the spin and the loop O ( n ) models and suggests that both these modelsreside in the same universality class. For this reason, it is natural to ask whether the predictionabout the absence of phase transition is valid for the loop O ( n ) model. Question 1.2.
Does the quantity on the right-hand side of (2) decay exponentially fast in thedistance between u and v , uniformly in the domain H , whenever n > and x > ? In this article, we partially answer this question. In Theorem 1.5 below, we show that for allsufficiently large n and any x >
0, the quantity on the right-hand side of (2) decays exponentiallyfast for a large class of domains H . The theorem is a consequence of a more detailed understandingof the loop O ( n ) model. We show that for small x the model is in a dilute, disordered phase, wherethe sampled loop configuration is rather sparse and the probability of seeing long loops surroundinga given vertex decays exponentially in the length (see Figure ). For large x , the same exponentialdecay holds but for a different reason. There, the model is in a dense, ordered phase, which is aperturbation of a periodic ground state. In the ground state all loops have length 6 and a typicalperturbation does not make them significantly longer (see Figure ). The x = ∞ Model.
We shall also consider the limit of the loop O ( n ) model as the edge weight x tends to infinity. This means restricting the model to ‘optimally packed loop configurations’, i.e.,loop configurations having the maximum possible number of edges. Definition 1.3.
Let H be a domain and let ξ be a loop configuration. For n > , the loop O ( n ) measure on H with edge weight x = ∞ and boundary conditions ξ is the probability measure on HUGO DUMINIL-COPIN, RON PELED, WOJCIECH SAMOTIJ, AND YINON SPINKA
Figure 1.
On the left, a loop configuration. On the right, a proper 3-coloring ofthe triangular lattice T (the dual of the hexagonal lattice H ), inducing a partitionof T into three color classes T , T , and T . The 0-phase ground state ω isthe (fully-packed) loop configuration consisting of trivial loops around each hexagonin T . LoopConf ( H, ξ ) defined by P ξH,n, ∞ ( ω ) := lim x →∞ P ξH,n,x ( ω ) = n LH ( ω ) Z ξH,n, ∞ if o H ( ω ) = o H,ξ otherwise , ω ∈ LoopConf ( H, ξ ) , where o H,ξ := max { o H ( ω ) : ω ∈ LoopConf ( H, ξ ) } and Z ξH,n, ∞ is the unique constant making P ξH,n, ∞ a probability measure. We note that if a loop configuration ω ∈ LoopConf ( H, ξ ) is fully packed , i.e., every vertex in V ( H )has degree 2, then ω is optimally packed, i.e., o H ( ω ) = o H,ξ .Before concluding this section, let us mention that the loop O ( n ) model with n ≤ Results.
In order to state our main results, we need several more definitions (see Figure fortheir illustration). We consider the triangular lattice T := (0 , Z +( √ , Z , and view the hexagonallattice H as its dual lattice, obtained by placing a vertex at the center of every face (triangle) of T ,so that each edge e of H corresponds to the unique edge e ∗ of T which intersects e . Since verticesof T are identified with faces of H , they will be called hexagons instead of vertices. We will also saythat a vertex or an edge of H borders a hexagon if it borders the corresponding face of H .There are exactly 6 proper colorings of T with the colors { , , } . For the rest of the paper, wefix an arbitrary proper coloring and let T c be the set of hexagons colored by c , c ∈ { , , } . A trivialloop is a loop of length exactly 6. Define the c -phase ground state ω c gnd to be the (fully-packed)loop configuration consisting of all the trivial loops surrounding hexagons in T c . We shall say thata domain H is of type c , c ∈ { , , } , if every edge { u, v } ∈ ω c gnd satisfies either u, v ∈ V ( H ) or u, v / ∈ V ( H ). Equivalently, H is of type c if and only if LoopConf ( H, ∅ ) = { ω ∩ E ( H ) : ω ∈ LoopConf ( H, ω c gnd ) } . (3)Finally, we shall say that a loop surrounds a vertex u of H if any infinite simple path in H starting at u intersects a vertex of this loop. In particular, if a loop passes through a vertex then it surroundsit as well. Theorem 1.4.
There exist n , α > such that for any n ≥ n and x ∈ (0 , ∞ ] the following holds.For any c ∈ { , , } , any domain H of type c , any u ∈ V ( H ) and any integer k > , we have P ∅ H,n,x ( there exists a loop of length k surrounding u ) ≤ n − αk . XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n (a) n = 8 and x = 0 .
5. Theorem 1.6 shows thatthe limiting measure is unique for domains with va-cant boundary conditions when x is small. (b) n = 8 and x = 2. Theorem 1.8 shows that typicalconfigurations are small perturbations of the groundstate for large n and x . Figure 2.
Two samples of random loop configurations with large n . Configurationsare on a 60 ×
45 domain of type 0 and are sampled via Glauber dynamics for 100million iterations started from the empty configuration.As follows from Theorem 1.8 below, when n and nx are sufficiently large, it is likely that u iscontained in a trivial loop. Thus, the assumption that k > Theorem 1.5.
There exist n , α > such that for any n ≥ n and any x > the following holds.For any c ∈ { , , } , any domain H of type c and any distinct non-adjacent u, v ∈ V ( H ) , we have (cid:88) λ ∈ LoopConf ( H, ∅ ,u,v ) x o H ( λ ) n L (cid:48) H ( λ ) J ( λ ) (cid:88) ω ∈ LoopConf ( H, ∅ ) x o H ( ω ) n L H ( ω ) ≤ x · n − α d H ( u,v ) , where d H ( u, v ) is the graph distance in H between u and v . Our techniques provide additional information on the (infinite-volume) Gibbs measures of theloop O ( n ) model. We recall the standard definition: a probability measure P on the set of loopconfigurations on H (viewed as a subset of { , } E ( H ) ) is a Gibbs measure for the loop O ( n ) modelwith edge weight x if for any domain H and P -almost every loop configuration ξ , the distributionof the configuration ω , conditioned that ω ∈ LoopConf ( H, ξ ), is given by P ξH,n,x .For small parameter x , under vacant boundary conditions, the model is in a dilute, disorderedphase, where loops are rare and tend to be short; see Figure . This is relatively simple to showand is proved in Corollary 3.2. A consequence of this fact is the existence of a unique limiting Gibbsmeasure when exhausting the hexagonal lattice H via domains with vacant boundary conditions. Theorem 1.6.
There exists c > such that for any n > and < x ≤ c satisfying nx ≤ c the following holds. Let H k be an increasing sequence of domains satisfying ∪ k H k = H . Then themeasures P ∅ H k ,n,x converge (weakly) as k → ∞ to an infinite-volume Gibbs measure P H ,n,x which issupported on loop configurations with no infinite paths. HUGO DUMINIL-COPIN, RON PELED, WOJCIECH SAMOTIJ, AND YINON SPINKA
It follows that the limiting measure P H ,n,x does not depend on the specific choice of exhaustingsequence ( H k ) as one may interleave two such sequences to obtain another convergent sequence.Consequently, it also follows that P H ,n,x is invariant under automorphisms of H . Our proofs applyalso when one allows H k to be arbitrary finite subgraphs of H rather than domains, but we donot state this explicitly as our work is mostly concerned with domains. The restriction to vacantboundary conditions is, however, essential for our proofs with the difficulty stemming from the factthat non-vacant boundary conditions may force the existence of long paths in the configuration (seeFigure ). Still, it may be that there is a unique Gibbs measure in this regime of small x and weprovide a discussion of this in Section 4.For large parameter x and large n , the situation changes dramatically. Here, we obtain thatthe model is in a dense, ordered phase, where, under the ω c gnd boundary conditions, a typicalconfiguration is a perturbation of that ground state. As a consequence of this structure, the modelhas at least three different limiting Gibbs measures in this regime of n and x . We state this preciselyin the following theorem. To lighten the notation, we write P c H,n,x for the loop O ( n ) measure on H with boundary conditions ω c gnd . Theorem 1.7.
There exists
C > such that for any n ≥ C and any x ∈ (0 , ∞ ] satisfying nx ≥ C the following holds. Let H k be an increasing sequence of domains satisfying ∪ k H k = H . Then, forevery c ∈ { , , } , the measures P c H k ,n,x converge (weakly) as k → ∞ to an infinite-volume Gibbsmeasure P c H ,n,x which is supported on loop configurations with no infinite paths. Furthermore, noone of the limiting measures is a convex combination of the other two. Similarly to before, it follows that, for each c ∈ { , , } , the limiting measure P c H ,n,x does notdepend on the specific choice of exhausting sequence ( H k ) and that P c H ,n,x is invariant under au-tomorphisms preserving the set T c . However, as these measures are distinct for different c , theyare not invariant under all automorphisms. In particular, if each H k is of type c , by (3), we havethat P ∅ H k ,n,x also converges to P c H ,n,x , in contrast to the behavior obtained in Theorem 1.6 for small x . It would be interesting to determine whether every infinite-volume Gibbs measure is a convexcombination of these three measures, i.e., whether these are the only extremal Gibbs measures (seealso Section 4). As we remark at the end of the section, this is not the case for x = ∞ .As mentioned above, in the ordered regime (large x and n ), a typical configuration drawn from P c H,n,x is a perturbation of the c -phase ground state ω c gnd (see Figure ). This is made precise inthe following theorem, which we state for the c = 0 phase for concreteness of our definitions. Inorder to measure how close ω and a typical loop configuration are, we introduce the notion ofa breakup . Fix a domain H and let ω ∈ LoopConf ( H, ω ) be a loop configuration. Let A ( ω ) bethe set of vertices of H belonging to trivial loops surrounding hexagons in T and let B ( ω ) be theunique infinite connected component of A ( ω ). For u ∈ H , define the breakup C ( ω, u ) of u to be theconnected component of H \ B ( ω ) containing u , setting C ( ω, u ) = ∅ if u ∈ B ( ω ). We also define ∂ C ( ω, u ) to be the internal vertex boundary of C ( ω, u ), i.e., the set of vertices in C ( ω, u ) adjacent toa vertex not in C ( ω, u ) (thus in B ( ω )). We remark that C ( ω, u ) need not be contained in H , thoughit cannot extend significantly beyond it in the sense that it is contained in any domain of type 0containing H . Theorem 1.8.
There exists c > such that for any n > , any x ∈ (0 , ∞ ] , any domain H , any u ∈ V ( H ) and any positive integer k , we have P H,n,x ( | ∂ C ( ω, u ) | ≥ k ) ≤ ( cn · min { x , } ) − k/ . One should note that the above theorem contains the implicit assumption that n ≥ C and nx ≥ C , as otherwise the statement is trivial.In this work, we mainly study the loop O ( n ) model with either vacant or ground state boundaryconditions. To obtain a complete picture regarding the possible Gibbs measures, one must also XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n (a) Domains for which there exists a single fully-packedloop configuration (with vacant boundary conditions). Us-ing such domains, one may obtain many weak limits of theprobability measures P ∅ H,n, ∞ . (b) A domain with boundary conditions in-ducing a unique loop configuration with min-imal number of edges. Such domains give riseto a Gibbs measure for x = 0 which containsan infinite interface passing near the origin. Figure 3.
Constructing multiple Gibbs measures when x = 0 or x = ∞ throughsuitable domains and boundary conditions.study the model for general boundary conditions. As mentioned above, understanding the Gibbsmeasures in each regime of n and x , and in particular, determining the number of extremal Gibbsmeasures, is an interesting problem. Theorem 1.6 and Theorem 1.7 bring us closer to this goal,providing a partial answer in the regimes nx ≤ c and nx ≥ C , for large n . In this regard, one mayask what happens in the intermediate regime, i.e., when c < nx < C and n is large. For instance,one may ask whether or not there is a single transition curve, perhaps of the form nx = c (cid:48) . If indeedthis is the case, it would be interesting to investigate the number of extremal Gibbs measures onthis curve, determining whether there is a unique such Gibbs measure (as Theorem 1.6 suggests for nx ≤ c ), 3 such measures (as Theorem 1.7 suggests for nx ≥ C ), 4 such measures, or perhaps adifferent quantity (see also Section 4). Remark.
For x = 0 and x = ∞ , many other Gibbs measures can be constructed. For instance,for positive integers a and b , let H a,b be the “rectangle” of width 2 a + 1 and height b (measuredin hexagons) with the origin at the center, as in Figure (on the left). It is not hard to checkthat the configuration depicted in the figure is the unique fully-packed loop configuration (withvacant boundary conditions) inside H a,b . Thus, the probability measure P ∅ H a,b ,n, ∞ is supported ona single configuration. The measures P ∅ H a,b ,n, ∞ converge (as a, b → ∞ ) to a delta measure on theconfiguration of infinite vertical paths covering the entire lattice (which is a Gibbs measure of theloop O ( n ) model with edge weight ∞ ). By considering different domains, one may construct manymore examples of this nature (once again, see Figure ). One may also look at the limiting modelas x tends to 0, which corresponds to requiring the configuration to have the minimal number ofedges. For the vacant boundary conditions, the finite-volume measure is a Dirac measure on theempty configuration. Using alternative boundary conditions, one may construct several distinctGibbs measures (see, e.g., Figure ).1.2. Overview of the proof.
Our proofs make use of the following simple lemma.
Lemma 1.9.
Let p, q > and let E and F be two events in a discrete probability space. If thereexists a map T : E → F such that P ( T ( e )) ≥ p · P ( e ) for every e ∈ E , and | T − ( f ) | ≤ q for every f ∈ F , then P ( E ) ≤ qp · P ( F ) . HUGO DUMINIL-COPIN, RON PELED, WOJCIECH SAMOTIJ, AND YINON SPINKA
Proof.
We have p · P ( E ) ≤ (cid:88) e ∈ E P ( T ( e )) = (cid:88) e ∈ E (cid:88) f ∈ F P ( f ) { T ( e )= f } = (cid:88) f ∈ F | T − ( f ) | · P ( f ) ≤ q · P ( F ) . (cid:3) The results for small x are obtained via a fairly standard, and short, Peierls argument, by applyingthe above lemma to a map which removes loops. For details, we refer the reader to Section 3.1. Themain novelty of this work lies in the study of the loop O ( n ) model for large x .In the large x regime, the idea is to apply the above lemma to a suitably defined ‘repair map’.This map takes a configuration ω sampled with 0-phase ground state boundary conditions (or vacantboundary conditions in a domain of type 0) and having a large breakup and returns a ‘repaired’configuration in which the breakup is significantly reduced. The map operates by identifying regionsin which the configuration resembles one of the three ground states. Regions resembling the ω state are ‘shifted down’ by one hexagon to resemble ω and similarly regions resembling ω are‘shifted up’ by one hexagon to resemble ω . Regions resembling the ω state are left untouched.Regions which do not resemble any of the ground states are completely replaced by trivial loops fromthe ω state. We show that this yields a new loop configuration, compatible with the boundaryconditions, and having much higher probability. To finish using Lemma 1.9, we further show thatthe number of preimages of a given loop configuration is exponentially smaller than the probabilitygain. This yields the main lemma of our paper, Lemma 2.10, from which our results for large x are later deduced. The repair map is illustrated in Figure and is formally defined in Section 2.3following the definitions of ‘flowers’, ‘gardens’ and ‘clusters’ which we require to make precise thenotion of resembling a ground state.1.3. Graph notation.
Throughout this paper, given a graph G , we shall denote its vertex andedge sets by V ( G ) and E ( G ), respectively. If u, v ∈ V ( G ) are such that { u, v } ∈ E ( G ), we say that u and v are adjacent (or neighbors ) in G and we drop the dependence on G if it is clear from thecontext. For a vertex u and an edge e such that u ∈ e , we say that e is incident to u and that u isan endpoint of e . For A ⊂ V ( G ), we define its (vertex) boundary ∂A by ∂A := (cid:8) u ∈ A : { u, v } ∈ E ( G ) for some v (cid:54)∈ A (cid:9) . The following is a standard lemma which gives a bound on the number of connected inducedsubgraphs of a graph.
Lemma 1.10 ([5, Chapter 45]) . Let G be a graph with maximum degree d ≥ . The number ofconnected subsets of V ( G ) containing a given vertex and k other vertices is at most ( e ( d − k . Organization of the article.
The rest of the article is structured as follows. Section 2introduces the repair map and proves the main lemma, Lemma 2.10. In Section 3, we derive ourtheorems. The statements regarding large x are deduced from the main lemma whereas the partspertaining to small x , being simpler, are obtained directly. In Section 4, we discuss several directionsfor future research.1.5. Acknowledgements.
We are grateful to two anonymous referees whose comments helped toimprove the exposition and elucidate the relation of the results with the existing literature.2.
Flowers, gardens and the repair map
This section is devoted to the formulation and proof of the main lemma, Lemma 2.10. We start bystating a few definitions in Section 2.1. In particular, we introduce the notions of a circuit , c -flower , c -garden and c -cluster , and gather some easy general facts about these objects. The main lemma isstated in Section 2.2 and the remaining sections are devoted to its proof. Section 2.3 introduces therepair map, which will play the role of T in Lemma 1.9. Section 2.4 compares the probability of aconfiguration and its image under the repair map (which corresponds to estimating p in Lemma 1.9). XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n Figure 4.
A garden. The dashed line denotes a vacant circuit σ ⊂ T \ T c , where c ∈ { , , } . The edges inside σ , along with the edges crossing σ , then comprise a c -garden of ω , since every hexagon in T c ∩ ∂ Int hex ( σ ) is surrounded by a trivial loop.Section 2.5 gathers the last ingredients (mainly an estimate for the number of possible preimagesunder the repair map, which corresponds to bounding q in Lemma 1.9) to conclude the proof ofLemma 2.10.2.1. Definitions and gardening. A circuit is a simple closed path in T , which may be viewed asa sequence of hexagons γ = ( γ , . . . , γ m ), m ≥
3, satisfying the following two properties: • γ m = γ and γ i (cid:54) = γ j for every 0 ≤ i < j < m , • γ i and γ i +1 are neighbors (in T ) for every 0 ≤ i < m .Define γ ∗ to be the set of edges { γ i , γ i +1 } ∗ ∈ E ( H ) for 0 ≤ i < m .We proceed with three standard geometric facts regarding circuits and domains. For completeness,these facts are proved in Appendix B. The first two facts constitute a discrete version of the Jordancurve theorem. Fact 2.1. If γ is a circuit then the removal of γ ∗ splits H into exactly two connected components,one of which is infinite, denoted by Ext( γ ) , and one of which is finite, denoted by Int( γ ) . Moreover,each of these are induced subgraphs of H . Let γ be a circuit. We denote the vertex sets and edge sets of Int( γ ) , Ext( γ ) by Int V ( γ ) , Ext V ( γ )and Int E ( γ ) , Ext E ( γ ), respectively. Note that { Int V ( γ ) , Ext V ( γ ) } is a partition of V ( H ) and that { Int E ( γ ) , Ext E ( γ ) , γ ∗ } is a partition of E ( H ). We also define Int hex ( γ ) to be the set of faces ofInt( γ ), i.e., the set of hexagons z ∈ T having all their six bordering vertices in Int V ( γ ). Since Int( γ )is induced, this is equivalent to having all six bordering edges in Int E ( γ ).Note that, by Fact 2.1, Int( γ ) is a domain. The converse is also true. Fact 2.2.
Circuits are in one-to-one correspondence with domains via γ ↔ Int( γ ) . Hence, every domain H may be written as H = Int( γ ) for some circuit γ . Recalling the definitionfrom Section 1.1 of a domain of type c ∈ { , , } , one should also note that H is of type c if andonly if γ ⊂ T \ T c . Fact 2.3.
Let σ and σ (cid:48) be two circuits such that σ ∗ ∩ ( σ (cid:48) ) ∗ (cid:54) = ∅ or Int V ( σ ) ∩ Int V ( σ (cid:48) ) (cid:54) = ∅ . Thenthere exists a circuit γ ⊂ σ ∪ σ (cid:48) such that γ ∗ ⊂ σ ∗ ∪ ( σ (cid:48) ) ∗ and Int( σ ) ∪ Int( σ (cid:48) ) ⊂ Int( γ ) . Definition 2.4 ( c -flower, c -garden, vacant circuit; see Figure ) . Let c ∈ { , , } and let ω be aloop configuration. A hexagon z ∈ T c is a c -flower of ω if it is surrounded by a trivial loop in ω . Asubset E ⊂ E ( H ) is a c -garden of ω if there exists a circuit σ ⊂ T \ T c such that E = Int E ( σ ) ∪ σ ∗ and every z ∈ T c ∩ ∂ Int hex ( σ ) is a c -flower of ω . In this case, we denote σ ( E ) := σ . A circuit σ is vacant in ω if ω ∩ σ ∗ = ∅ . We say that E ⊂ E ( H ) is a garden of ω if it is a c -garden of ω for some c ∈ { , , } . We stressthe fact that a garden is a subset of the edges of H . We continue with several simple properties ofcircuits, gardens and loop configurations which will be used throughout the paper. Lemma 2.5.
Let ω and ω (cid:48) be two loop configurations.(a) If σ is a vacant circuit in ω then ω ∩ Int E ( σ ) and ω ∩ Ext E ( σ ) are loop configurations.(b) If E is a garden of ω then σ ( E ) is a vacant circuit in ω .(c) If E is a garden of ω then ω ∩ E and ω \ E are loop configurations.(d) If ω and ω (cid:48) are disjoint then ω ∪ ω (cid:48) is a loop configuration.(e) If ω (cid:48) is contained in ω then ω \ ω (cid:48) is a loop configuration.Proof. To see (a), let σ be a vacant circuit in ω . Since any path between Int( σ ) and Ext( σ ) intersects σ ∗ , and since ω ∩ σ ∗ = ∅ , every loop of ω is contained in either Int( σ ) or Ext( σ ), and thus, (a) follows.We now show (b). Let E be a c -garden of ω , c ∈ { , , } , and let σ := σ ( E ). One of the endpointsof every edge e ∈ σ ∗ must border a hexagon in T c ∩ ∂ Int hex ( σ ). By the definition of a c -garden, thishexagon is a c -flower, and hence, e cannot belong to ω . Thus, σ is vacant in ω .In light of (a) and (b), (c) is immediate.To establish (d), it suffices to show that no vertex has degree 3 in ω (cid:48) ∪ ω . Indeed, if a vertexhas degree 3 then one of the edges incident to it must be contained in both ω and ω (cid:48) , which is acontradiction.Finally, the last statement is straightforward. (cid:3) Lemma 2.6.
Let c ∈ { , , } , let σ ⊂ T \ T c be a circuit, let z ∈ T c be a hexagon and let V ( z ) denote the six vertices in H bordering z . Then z ∈ Int hex ( σ ) ⇐⇒ V ( z ) ∩ Int V ( σ ) (cid:54) = ∅ . Proof.
Recall that, by definition, z ∈ Int hex ( σ ) if and only if V ( z ) ⊂ Int V ( σ ). Thus, it suffices tocheck that if v ∈ V ( z ) ∩ Int V ( σ ) and u ∈ V ( z ) is adjacent to v then u ∈ Int V ( σ ). Indeed this is thecase, as otherwise, { u, v } ∈ σ ∗ and z ∈ σ , which contradicts the assumption that σ ⊂ T \ T c . (cid:3) We proceed to discuss disjointness and containment properties of gardens.
Lemma 2.7.
Let ω be a loop configuration and let E and E be two c -gardens of ω for some c ∈ { , , } . If there exists a vertex which is the endpoint of an edge in E and an edge in E , then E ∪ E is contained in a c -garden of ω .Proof. Denote σ := σ ( E ) and σ := σ ( E ). Let us first show that necessarily Int V ( σ ) ∩ Int V ( σ ) (cid:54) = ∅ or σ ∗ ∩ σ ∗ (cid:54) = ∅ . To this end, let v, u, w ∈ V ( H ) be such that { v, u } ∈ E and { v, w } ∈ E . If v ∈ Int V ( σ ) ∩ Int V ( σ ) then we are done. Otherwise, suppose without loss of generality that v ∈ Ext V ( σ ) so that u ∈ Int V ( σ ). If also v ∈ Ext V ( σ ) then necessarily w = u and w ∈ Int V ( σ )as σ , σ ⊂ T \ T c . If instead v ∈ Int V ( σ ) then either u ∈ Int V ( σ ) or { v, u } ∈ σ ∗ ∩ σ ∗ .By Fact 2.3, there exists a circuit γ such that γ ∗ ⊂ σ ∗ ∪ σ ∗ and Int( σ ) ∪ Int( σ ) ⊂ Int( γ ). Inparticular, E ∪ E ⊂ E , where E := Int E ( γ ) ∪ γ ∗ . It remains to show that E is a c -garden of ω .Since, by Lemma 2.6, T c ∩ ∂ Int hex ( γ ) ⊂ ∂ Int hex ( σ ) ∪ ∂ Int hex ( σ ), this follows from the assumptionthat E and E are c -gardens of ω . (cid:3) Lemma 2.8.
Let ω be a loop configuration, let E be a c -garden of ω and let E be a c -garden of ω with c , c ∈ { , , } distinct. Then, either E ⊂ E , E ⊂ E or E ∩ E = ∅ .Proof. Assume without loss of generality that c = 0, c = 1 and that E ∩ E (cid:54) = ∅ . Denote σ := σ ( E ) ⊂ T \ T and σ := σ ( E ) ⊂ T \ T . Consider an infinite path in H beginning with someedge of E ∩ E and let e ∈ E ( H ) be the first edge on this path that is not in Int E ( σ ) ∩ Int E ( σ )(maybe the first edge itself). We may assume without loss of generality that e / ∈ Int E ( σ ). Thus, e ∈ σ ∗ , and, therefore, e is bordered by a hexagon z ∈ T and a hexagon in T . Since e is also in XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n Figure 5.
A loop configuration ω ∈ LoopConf ( H, ω ). The 0-clusters are denotedin green, the 1-clusters in red and the 2-clusters in blue; all taken with respect to thecircuit surrounding the large unshaded domain. E , z belongs to Int hex ( σ ), by Lemma 2.6. Now, if σ ⊂ Int hex ( σ ) then E ⊂ E , by Fact 2.1.Otherwise, there exists { y, y (cid:48) } ∈ σ ∗ such that y ∈ Int hex ( σ ) and y (cid:48) / ∈ Int hex ( σ ). In particular, y (cid:48) must be in σ ∩ σ ⊂ T , so that y must be in T . Since y is in ∂ Int hex ( σ ), it must be a 1-flower of ω . But since y is on σ , it must also be adjacent to a 0-flower of ω , which is a contradiction. (cid:3) Definition 2.9 ( c -cluster, c -cluster inside γ ) . Let c ∈ { , , } and let ω be a loop configuration. Asubset E ⊂ E ( H ) is a c -cluster of ω if it is a c -garden of ω and it is not contained in any othergarden of ω . Let γ be a vacant circuit in ω and note that ω ∩ Int E ( γ ) is a loop configuration byLemma 2.5a. A subset E ⊂ E ( H ) is a c -cluster of ω inside γ if it is a c -cluster of ω ∩ Int E ( γ ) . We say that E ⊂ E ( H ) is a cluster (inside γ ) if it is a c -cluster (inside γ ) for some c ∈ { , , } .Once again, note that a cluster (inside γ ) is a subset of edges of H . Evidently, a cluster of ω inside γ is also a garden of ω , but it is not necessarily a cluster of ω . The notion of c -cluster inside γ will be important in the definition of the repair map in Section 2.3. Note that, by Lemma 2.7 andLemma 2.8, any two distinct clusters of ω (inside γ ) are edge disjoint , (4)and, moreover, for any c ∈ { , , } ,the union of any two distinct c -clusters of ω (inside γ ) is a disconnected set of edges , (5)where a set of edges E is said to be connected if the graph whose vertex set is the set of endpointsof edges in E and whose edge set is E is connected. Note also, that by Fact 2.1,every cluster of ω (inside γ ) is a connected set of edges . (6) Statement of the main lemma.
We are now in a position to state the main lemma. For aloop configuration ω and a vacant circuit γ in ω , denote by V ( ω, γ ) the set of vertices v ∈ Int V ( γ )such that the three edges of H incident to v are not all contained in the same cluster of ω inside γ . One checks simply using Lemma 2.6 that a vertex v ∈ Int V ( γ ) satisfies v ∈ V ( ω, γ ) if and onlyif v is incident to an edge which is not in any cluster or each of its incident edges lies in a differentcluster.For a vacant circuit γ ⊂ T \ T , the set V ( ω, γ ) specifies the deviation in ω from the 0-phaseground state along the interior boundary of γ . Our main lemma shows that having a large deviationis exponentially unlikely. Lemma 2.10.
There exists an absolute constant c > such that for any n > , any x ∈ (0 , ∞ ] ,any domain H , any circuit γ ⊂ T \ T and any positive integer k , we have P H,n,x (cid:0) ∂ Int V ( γ ) ⊂ V ( ω, γ ) and | V ( ω, γ ) | ≥ k | γ vacant (cid:1) ≤ ( cn · min { x , } ) − k/ . The reader should first have in mind the simpler case of the lemma in which H = Int( γ ). Inthis case the boundary conditions may equivalently be taken to be vacant. The lemma is stated ingreater generality, allowing, in particular, for γ to leave the domain H , i.e., for Int( γ ) (cid:54)⊂ H . Thisadditional flexibility is used in the proofs of Theorem 1.7 and Theorem 1.8 to handle the case ofdomains without a type.One should note that Lemma 2.10 contains the implicit assumption that n ≥ n · min { x , } ≥ C ,as otherwise its statement is trivial.2.3. Definition of the repair map.
For the remainder of this section, we fix a circuit γ ⊂ T \ T and set H := Int( γ ). Consider a loop configuration ω such that γ is vacant in ω . The idea of therepair map is to modify ω as follows: • Edges in 1-clusters inside γ are shifted down “into the 0-phase”. • Edges in 2-clusters inside γ are shifted up “into the 0-phase”. • Edges in 0-clusters inside γ are left untouched. • The remaining edges which are not inside (the shifted) clusters, but are in the interior of γ (these edges will be called bad ), are overwritten to “match” the 0-phase ground state, ω .See Figure for an illustration of this map.In order to formalize this idea, we need a few definitions. A shift is a graph automorphism of T which maps every hexagon to one of its neighbors. We henceforth fix a shift ↑ which maps T to T (and hence, maps T to T and T to T ), and denote its inverse by ↓ . A shift naturallyinduces mappings on the set of vertices and the set of edges of H . We shall use the same symbols, ↑ and ↓ , to denote these mappings. Recall from Section 1.1 that T has a coordinate system givenby (0 , Z + ( √ , Z and that ( T , T , T ) are the color classes of an arbitrary proper 3-coloringof T . In our figures we make the choice that (0 , ∈ T and (0 , ∈ T so that ↑ is the map( a, b ) (cid:55)→ ( a, b + 2).For a loop configuration ω ∈ LoopConf ( H, ∅ ) and c ∈ { , , } , let E c ( ω ) ⊂ E ( H ) be the union ofall c -clusters of ω . Note that, since H = Int( γ ), for ω ∈ LoopConf ( H, ∅ ), the notions of a c -clusterand a c -cluster inside γ coincide. For ω ∈ LoopConf ( H, ∅ ), define also E bad ( ω ) := (Int E ( γ ) ∪ γ ∗ ) \ (cid:0) E ( ω ) ∪ E ( ω ) ↓ ∪ E ( ω ) ↑ (cid:1) , (7) E ( ω ) := (Int E ( γ ) ∪ γ ∗ ) \ (cid:0) E ( ω ) ∪ E ( ω ) ∪ E ( ω ) (cid:1) . (8)Note that, by (4), { E ( ω ) , E ( ω ) , E ( ω ) , E ( ω ) } is a partition of Int E ( γ ) ∪ γ ∗ . Thus, Lemma 2.5implies that ω ∩ E ( ω ), ω ∩ E ( ω ), ω ∩ E ( ω ) and ω ∩ E ( ω ) are pairwise disjoint loop configurations. (9)See Figure and Figure for an illustration of these notions. Finally, we define the repair map R γ : LoopConf ( H, ∅ ) → LoopConf ( H, ∅ ) XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n by R γ ( ω ) := (cid:0) ω ∩ E ( ω ) (cid:1) ∪ (cid:0) ω ∩ E ( ω ) (cid:1) ↓ ∪ (cid:0) ω ∩ E ( ω ) (cid:1) ↑ ∪ (cid:0) ω ∩ E bad ( ω ) (cid:1) . The fact that the mapping is well-defined, i.e., that R γ ( ω ) is indeed in LoopConf ( H, ∅ ), is notcompletely straightforward. This follows from the following proposition, together with the simpleproperty in Lemma 2.5d. Proposition 2.11.
Let ω ∈ LoopConf ( H, ∅ ) . Then ω ∩ E ( ω ) , ( ω ∩ E ( ω )) ↓ ∪ ( ω ∩ E ( ω )) ↑ and ω ∩ E bad ( ω ) are pairwise disjoint loop configurations in LoopConf ( H, ∅ ) . We require the following simple geometric lemma.
Lemma 2.12.
Let σ ⊂ T \ T and σ (cid:48) ⊂ T \ T be circuits.(a) If Int( σ (cid:48) ) ⊂ Int( σ ) then Int( σ (cid:48) ) ↓ ⊂ Int( σ ) .(b) If Int( σ (cid:48) ) ⊂ Ext( σ ) then Int( σ (cid:48) ) ↓ ⊂ Ext( σ ) .(c) If Int V ( σ (cid:48) ) ∩ Int V ( σ ) = ∅ then Int V ( σ (cid:48) ) ↓ ∩ Int V ( σ ) = ∅ .Proof. We first prove (a). The assumption that Int( σ (cid:48) ) ⊂ Int( σ ) implies that Int hex ( σ (cid:48) ) ⊂ Int hex ( σ ).By Lemma 2.6, any vertex v in Int V ( σ (cid:48) ) borders a hexagon in Int hex ( σ (cid:48) ). Thus, it suffices toshow that Int hex ( σ (cid:48) ) ↓ ⊂ Int hex ( σ ). Assume towards a contradiction that there exists a hexagon z ∈ Int hex ( σ (cid:48) ) such that z ↓ / ∈ Int hex ( σ ). In such case, by Fact 2.1, z ↓ must be in σ ∩ σ (cid:48) ⊂ T , andconsequently, z ∈ T . Therefore, as z ∈ Int hex ( σ (cid:48) ) and σ (cid:48) ⊂ T \ T , Lemma 2.6 implies that thethree neighbors of z in T belong to Int hex ( σ (cid:48) ) ⊂ Int hex ( σ ). Now, Lemma 2.6 implies that z ↓ hasthree neighbors in T ∩ Int hex ( σ ). In particular, the six vertices bordering z ↓ belong to Int V ( σ ),implying that z ↓ ∈ Int hex ( σ ), which is a contradiction.The proof of (b) is very similar to that of (a) and so we omit it.Finally, by Fact 2.1, (c) is equivalent to (b). (cid:3) Proof of Proposition 2.11.
For the sake of brevity, throughout the proof, we drop ω from the notationof the above sets and write E bad , E , E and E . Step 1: ω ∩ E , ( ω ∩ E ) ↓ ∪ ( ω ∩ E ) ↑ and ω ∩ E bad are contained in Int E ( γ ) . Since γ is vacant in both ω and ω , it follows that ω ∩ E and ω ∩ E bad are contained inInt E ( γ ). It remains to show that ( ω ∩ E ) ↓ and ( ω ∩ E ) ↑ are contained in Int E ( γ ). We showthis only for ( ω ∩ E ) ↓ , as the other case is symmetric. Let E be a 1-cluster of ω . We must showthat ( ω ∩ E ) ↓ ⊂ Int E ( γ ). Since, by Lemma 2.5b, ω ∩ E ⊂ Int E ( σ ( E )) ⊂ Int E ( γ ), this follows fromLemma 2.12a. Step 2: ω ∩ E , ( ω ∩ E ) ↓ ∪ ( ω ∩ E ) ↑ and ω ∩ E bad are pairwise disjoint. By definition, E bad (and therefore ω ∩ E bad ) is disjoint from the first two sets. It remains toshow that ω ∩ E is disjoint from ( ω ∩ E ) ↓ and ( ω ∩ E ) ↑ . We show this only for ω ∩ E and ( ω ∩ E ) ↓ ,as the other case is symmetric. Let E and E (cid:48) be 0- and 1-clusters of ω , respectively. We must showthat ( ω ∩ E ) ∩ ( ω ∩ E (cid:48) ) ↓ = ∅ . By Lemma 2.5b, ( ω ∩ E ) ∩ ( ω ∩ E (cid:48) ) ↓ ⊂ Int E ( σ ( E )) ∩ Int E ( σ ( E (cid:48) )) ↓ ,which is empty by (4) and Lemma 2.12c. Step 3: ω ∩ E , ( ω ∩ E ) ↓ ∪ ( ω ∩ E ) ↑ and ω ∩ E bad are loop configurations. We first show that ω ∩ E bad is a loop configuration. Observe that E ∪ ( E ) ↓ ∪ ( E ) ↑ is theunion of Int E ( σ ) ∪ σ ∗ for a collection of circuits σ ⊂ T \ T . Since every circuit σ ⊂ T \ T is vacantin ω , Lemma 2.5 implies that ω ∩ ( E ∪ ( E ) ↓ ∪ ( E ) ↑ ) is a loop configuration, and thus, alsothat ω ∩ E bad = ( ω \ ( E ∪ ( E ) ↓ ∪ ( E ) ↑ )) ∩ Int E ( γ ) is a loop configuration.Since ω ∩ E is a loop configuration, by (9), it remains only to check that ( ω ∩ E ) ↓ ∪ ( ω ∩ E ) ↑ is aloop configuration. In light of (9) and Lemma 2.5(d,e), it suffices to show that ( ω ∩ E ) ↓ ∩ ( ω ∩ E ) ↑ is a loop configuration. For convenience, we prove this separately in the next lemma. (cid:3) a) The breakup is found by exploring 0-flowers from theboundary. (b)
The clusters are found within the breakup. (c)
Bad edges are discarded. (d)
The clusters are shifted into the 0-phase. (e)
The empty area outside the shifted clusters is nowcompatible with the 0-phase ground state. (f)
Trivial loops are packed in the empty area outside theshifted clusters.
Figure 6.
An illustration of finding the breakup and applying the repair map in it. Theinitial loop configuration is modified step-by-step, resulting in a loop configuration withmany more loops and at least as many edges. Formal definitions are in Section 2.3.
XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n For a hexagon z ∈ T , we denote by E ( z ) the six edges bordering z . We call a hexagon z ∈ T double-clustered for ω if E ( z ↑ ) ⊂ E ( ω ) and E ( z ↓ ) ⊂ E ( ω ). Denote by dbl( ω ) the subset of allhexagons in Int hex ( γ ) that are double-clustered for ω . Lemma 2.13.
Let ω ∈ LoopConf ( H, ∅ ) . Then dbl( ω ) ⊂ T and ( ω ∩ E ( ω )) ↓ ∩ ( ω ∩ E ( ω )) ↑ consists solely of the trivial loops surrounding the hexagons in dbl( ω ) . That is, ( ω ∩ E ( ω )) ↓ ∩ ( ω ∩ E ( ω )) ↑ = (cid:91) z ∈ dbl( ω ) E ( z ) . Proof.
Let z ∈ dbl( ω ). Then z ↑ ∈ Int hex ( σ ( E )) and z ↓ ∈ Int hex ( σ ( E )), where E and E are1- and 2-clusters of ω , respectively. It follows from Lemma 2.6 and (4) that z ∈ T and that z / ∈ Int hex ( σ ( E )) ∪ Int hex ( σ ( E )). Thus, z ↑ is a 1-flower of ω and z ↓ is a 2-flower of ω . Inparticular, E ( z ) ⊂ ( ω ∩ E ( ω )) ↓ ∩ ( ω ∩ E ( ω )) ↑ .For the opposite containment, let e ∈ ( ω ∩ E ( ω )) ↓ ∩ ( ω ∩ E ( ω )) ↑ . Then e ↑ ∈ Int E ( σ ( E )) ∪ σ ( E ) ∗ and e ↓ ∈ Int E ( σ ( E )) ∪ σ ( E ) ∗ , where E and E are 1- and 2-clusters of ω , respectively. Since, byLemma 2.5b, σ ( E ) and σ ( E ) are vacant in ω , we have e ↑ ∈ Int E ( σ ( E )) and e ↓ ∈ Int E ( σ ( E )).In particular, both endpoints of e ↑ belong to Int V ( σ ( E )) and both endpoints of e ↓ belong toInt V ( σ ( E )). Therefore, by Lemma 2.6, e must border a hexagon z in T , and E ( z ↑ ) ⊂ E and E ( z ↓ ) ⊂ E . Thus, z ∈ dbl( ω ). (cid:3) The next lemma shows that certain boundary conditions are preserved by the repair map.
Lemma 2.14.
Let H (cid:48) be a domain and denote E := LoopConf ( H, ∅ ) ∩ LoopConf ( H (cid:48) , ω ∩ Int E ( γ )) .Then R γ ( E ) ⊂ E .Proof. Let ω ∈ E and denote ω (cid:48) := R γ ( ω ). Set F := Int E ( γ ) \ E ( H (cid:48) ) and note that E = { ˜ ω ∈ LoopConf ( H, ∅ ) : ˜ ω ∩ F = ω ∩ F } . In fact, one easily checks that E = { ˜ ω ∈ LoopConf ( H, ∅ ) : ω ∩ F ⊂ ˜ ω } . Thus, by Proposition 2.11, it suffices to show that ω ∩ F ⊂ ω (cid:48) .Let us first show that F is disjoint from E ( ω ) and E ( ω ). To this end, let e ∈ F and consideran infinite simple path in E ( H (cid:48) ) c starting from e . Observe that no vertex on this path borders a 1-or 2-flower of ω . On the other hand, by the definition of a cluster, if e belongs to a 1- or 2-clusterof ω , then any such path must have such a vertex. Hence, e / ∈ E ( ω ) ∪ E ( ω ).Towards showing that ω ∩ F ⊂ ω (cid:48) , let e ∈ ω ∩ F and note that e borders a hexagon z ∈ T .By Lemma 2.6, E ( z ) is contained in either E ( ω ), E ( ω ) ↓ , E ( ω ) ↑ or E bad ( ω ). In the first case, E ( z ) ⊂ ω ∩ E ( ω ) ⊂ ω (cid:48) . In the second case, z ↑ ∈ Int hex ( σ ( E )) for some 1-cluster E of ω . Since e / ∈ E ( ω ), we have z ↑ ∈ ∂ Int hex ( σ ( E )). Thus, z ↑ is a 1-flower of ω and E ( z ) ⊂ ( ω ∩ E ( ω )) ↓ ⊂ ω (cid:48) .The third case is similar to the second case. Finally, in the last case, E ( z ) ⊂ ω ∩ E bad ( ω ) ⊂ ω (cid:48) . (cid:3) Comparing the probabilities of R γ ( ω ) and ω . As in Section 2.3, we henceforth fix a circuit γ ⊂ T \ T and denote H := Int( γ ). Our goal now is to compare the probabilities of R γ ( ω ) and ω .Recall the definition of V ( ω, γ ) from Section 2.2. Denote by V (cid:48) ( ω, γ ) the vertices in V ( ω, γ ) whichare isolated in ω (i.e., which are incident to no edges in ω ). Proposition 2.15.
Let n ≥ , let x ∈ (0 , ∞ ] and let ω ∈ LoopConf ( H, ∅ ) . Then P ∅ H,n,x ( R γ ( ω )) ≥ n | V ( ω,γ ) | + | V (cid:48) ( ω,γ ) | · x | V (cid:48) ( ω,γ ) | · P ∅ H,n,x ( ω ) . In particular, if nx ≥ then P ∅ H,n,x ( R γ ( ω )) ≥ ( n · min { x , } ) | V ( ω,γ ) | · (max { x, } ) | V (cid:48) ( ω,γ ) | · P ∅ H,n,x ( ω ) . The proof of Proposition 2.15 is based on showing that applying the repair map can only increasethe number of loops and edges and estimating carefully the amounts by which they increase.
We begin with two preliminary lemmas. Denote by V bad ( ω ) the subset of Int V ( γ ) composed ofendpoints of edges in E bad ( ω ). Recall the definition of dbl( ω ) just prior to Lemma 2.13. Lemma 2.16.
For any ω ∈ LoopConf ( H, ∅ ) , we have | V bad ( ω ) | = | V ( ω, γ ) | + 6 · | dbl( ω ) | . Proof.
As before, set E c := E c ( ω ) for c ∈ { , , } . Let U := Int V ( γ ) \ V ( ω, γ ) be the set of verticeswhose three incident edges are contained in one of the sets E , E or E . Let U (cid:48) := Int V ( γ ) \ V bad ( ω )be the set of vertices whose three incident edges are contained in one of the sets E , ( E ) ↓ or ( E ) ↑ .The lemma will follow if we show that | U | − | U (cid:48) | = 6 · | dbl( ω ) | .For E ⊂ E ( H ), denote by Int( E ) the set of vertices whose 3 incident edges belong to E . Then U = Int( E ) ∪ Int( E ) ∪ Int( E ) ,U (cid:48) = Int( E ) ∪ Int( E ) ↓ ∪ Int( E ) ↑ . (10)We now show that Int( E ) ∩ Int( E ) ↓ = ∅ and Int( E ) ∩ Int( E ) ↑ = ∅ . (11)Note that, for a garden E , we have Int( E ) = Int V ( σ ( E )). Thus, it follows from (4) and Lemma 2.12cthat if E and E (cid:48) are 0- and 1-clusters of ω , respectively, then Int( E ) ∩ Int( E (cid:48) ) ↓ = ∅ . On the otherhand, Int( E c ) = ∪ Int( E ) over all c -clusters E of ω in γ , as follows from (5) and (6). We thereforeconclude that Int( E ) ∩ Int( E ) ↓ = ∅ . By symmetry, we also have Int( E ) ∩ Int( E ) ↑ = ∅ .Using the inclusion-exclusion principle, we obtain | U (cid:48) | = | Int( E ) | + | Int( E ) ↓ | + | Int( E ) ↑ | − | Int( E ) ↓ ∩ Int( E ) ↑ | by (10) and (11) = | Int( E ) | + | Int( E ) | + | Int( E ) | − | Int( E ) ↓ ∩ Int( E ) ↑ | = | U | − | Int( E ) ↓ ∩ Int( E ) ↑ | . by (10) and (4) Finally, observe that, by Lemma 2.6, Int( E ) ↓ ∩ Int( E ) ↑ is precisely the set of vertices that borderthe hexagons in dbl( ω ) and that each such vertex is incident to a unique double-clustered hexagon(since dbl( ω ) ⊂ T , by Lemma 2.13). Consequently, | Int( E ) ↓ ∩ Int( E ) ↑ | = 6 · | dbl( ω ) | . (cid:3) For our next lemma, we require the following definition. A functional on loops is a map φ thatassigns a real number to each loop in H . We say that φ is ↑ -invariant if φ ( L ↑ ) = φ ( L ) for everyloop L and φ ( L ) = φ ( L (cid:48) ) for any two trivial loops L and L (cid:48) . Given such a functional, we extend φ to finite loop configurations ω by summing over all the loops, i.e., by setting φ ( ω ) := (cid:88) loops L in ω φ ( L ) . Recall the definition of E ( ω ) from (8) and the repair map from Section 2.3. Let TrivLoop ⊂ H denote a trivial loop. Lemma 2.17.
For any ω ∈ LoopConf ( H, ∅ ) and any ↑ -invariant functional φ on loops, we have φ ( R γ ( ω )) − φ ( ω ) = φ ( TrivLoop ) · | V ( ω,γ ) | − φ ( ω ∩ E ( ω )) . Proof.
As before, set E c := E c ( ω ) for c ∈ { , , } and E bad := E bad ( ω ). Recall from Proposi-tion 2.11 that each loop of R γ ( ω ) belongs to one of the following pairwise disjoint loop configura-tions: ω ∩ E , ω ∩ E bad , or ( ω ∩ E ) ↓ ∪ ( ω ∩ E ) ↑ . Thus, the definition of a functional impliesthat φ ( R γ ( ω )) = φ ( ω ∩ E ) + φ ( ω ∩ E bad ) + φ (cid:0) ( ω ∩ E ) ↓ ∪ ( ω ∩ E ) ↑ (cid:1) . (12) XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n We claim that ω ∩ E bad consists of | V bad ( ω ) | / ω ∩ E bad is a loop con-figuration and ω is a fully-packed loop configuration (i.e., every vertex has degree 2) containingonly trivial loops, it suffices to show that each vertex in V bad ( ω ) is incident to at least two edges in E bad . We may write E bad = (Int E ( γ ) ∪ γ ∗ ) \ (cid:91) i (Int E ( σ i ) ∪ σ ∗ i ) = (cid:92) i Ext E ( σ i ) \ Ext E ( γ )for some circuits σ i ⊂ T \ T . Let v ∈ V bad ( ω ) and let z be the hexagon in T which v borders. ByLemma 2.6, the six edges bordering z must belong to Int E ( γ ) and to Ext E ( σ i ) for each i . Hence,they belong to E bad , and, in particular, two edges incident to v belong to E bad , as required.Thus, the ↑ -invariance of φ implies φ ( ω ∩ E bad ) = φ ( TrivLoop ) · | V bad ( ω ) | / . (13)By Lemma 2.13, the inclusion-exclusion principle and the ↑ -invariance of φ , we have that φ (cid:0) ( ω ∩ E ) ↓ ∪ ( ω ∩ E ) ↑ (cid:1) = φ (( ω ∩ E ) ↓ ) + φ (( ω ∩ E ) ↑ ) − φ (cid:0) ( ω ∩ E ) ↓ ∩ ( ω ∩ E ) ↑ (cid:1) = φ ( ω ∩ E ) + φ ( ω ∩ E ) − φ ( TrivLoop ) · | dbl( ω ) | . (14)Using identities (12), (13), (14) and Lemma 2.16, we obtain φ ( R γ ( ω )) = φ ( ω ∩ E ) + φ ( ω ∩ E ) + φ ( ω ∩ E ) + φ ( TrivLoop ) · | V ( ω, γ ) | / . Finally, by (9), φ ( ω ) = φ ( ω ∩ E ) + φ ( ω ∩ E ) + φ ( ω ∩ E ) + φ ( ω ∩ E ( ω )) , and the lemma follows by subtracting the last two displayed equations. (cid:3) Proof of Proposition 2.15.
Fix a loop configuration ω ∈ LoopConf ( H, ∅ ). Lemma 2.17 applied tothe ↑ -invariant functionals φ and φ defined by φ ( L ) := | E ( L ) | and φ ( L ) := 1 for every loop L implies (respectively) that∆ o := o H ( R γ ( ω )) − o H ( ω ) = | V ( ω, γ ) | − | ω ∩ E ( ω ) | , (15)∆ L := L H ( R γ ( ω )) − L H ( ω ) = | V ( ω, γ ) | / − L H ( ω ∩ E ( ω )) . (16)Since every trivial loop of ω is contained in a cluster, there are no trivial loops of ω in E ( ω ). Hence,as any non-trivial loop contains at least 10 edges, L H ( ω ∩ E ( ω )) ≤ | ω ∩ E ( ω ) | / . Furthermore, the simple observation that V ( ω, γ ) \ V (cid:48) ( ω, γ ) is precisely the set of endpoints of edgesin ω ∩ E ( ω ), and the fact that ω ∩ E ( ω ) is a loop configuration, by (9), imply that | ω ∩ E ( ω ) | = | V ( ω, γ ) \ V (cid:48) ( ω, γ ) | . Substituting these in (15) and (16), we obtain∆ o = | V (cid:48) ( ω, γ ) | and ∆ L ≥ | V ( ω,γ ) | + | V (cid:48) ( ω,γ ) | . Therefore, as n ≥ P ∅ H,n,x ( R γ ( ω )) P ∅ H,n,x ( ω ) = x o H ( R γ ( ω )) · n L H ( R γ ( ω )) x o H ( ω ) · n L H ( ω ) = x ∆ o · n ∆ L ≥ x | V (cid:48) ( ω,γ ) | · n | V ( ω,γ ) | + | V (cid:48) ( ω,γ ) | . (cid:3) u (cid:48) v (cid:48) u vz Figure 7.
If a circuit γ lies in T \ T then any three consecutive hexagons on γ arein the depicted constellation up to rotation and reflection (with γ denoted by thedotted line). The set of vertices in ∂ Int V ( γ ) bordering the hexagon z is then eitherthe set { u, v } or the set { u (cid:48) , v (cid:48) } , and in both cases, constitutes an edge of H × . Thesame is true for ∂ Ext V ( γ ).2.5. Proof of the main lemma.
In this section, we prove Lemma 2.10. Recall the definition of V ( ω, γ ) from Section 2.2. Let us start with two technical lemmas regarding the connectedness of V ( ω, γ ). Let H × be the graph obtained from H by adding an edge between each pair of oppositevertices of every hexagon, so that H × is a 6-regular non-planar graph. Lemma 2.18.
Let γ ⊂ T \ T be a circuit. Then ∂ Int V ( γ ) and ∂ Ext V ( γ ) are connected in H × .Proof. Suppose γ = ( z , . . . , z m ). Set U to be either ∂ Int V ( γ ) or ∂ Ext V ( γ ) and let U i be the set ofvertices in U which border the hexagon z i . The connectivity of U in H × is a consequence of thefollowing statements:(a) U = ∪ i U i .(b) U i ∩ U i +1 (cid:54) = ∅ for 0 ≤ i < m .(c) U i is connected in H × for all i .The first and second properties follow from Fact 2.1. For the third property note that the onlyconstellation up to rotation and reflection of three consecutive hexagons z i − , z i , z i +1 ∈ T \ T (where the indices are taken modulo m ) on γ is as depicted in Figure , so that the set U i has size2 and constitutes an edge in H × . (cid:3) Lemma 2.19.
Let ω be a loop configuration and let γ ⊂ T \ T be a vacant circuit in ω . If ∂ Int V ( γ ) ⊂ V ( ω, γ ) then V ( ω, γ ) is connected in H × .Proof. Let E , . . . , E m denote the clusters of ω inside γ and write σ i := σ ( E i ). The connectivity of V ( ω, γ ) in H × is a consequence of the following statements:(a) V ( ω, γ ) = Int V ( γ ) \ ∪ i Int V ( σ i ).(b) Int V ( γ ) is connected in H .(c) ∂ Ext V ( σ i ) is connected in H × for all i .(d) ∂ Ext V ( σ i ) ⊂ V ( ω, γ ) for all i .The first property follows from the definition of V ( ω, γ ), the second from Fact 2.1 and the third fromLemma 2.18 (and symmetry). For the fourth property, note that ∂ Ext V ( σ i ) ∩ Int V ( γ ) ⊂ V ( ω, γ )by (4), and ∂ Ext V ( σ i ) ⊂ Int V ( γ ) by the assumption that ∂ Int V ( γ ) ⊂ V ( ω, γ ). (cid:3) Lemma 2.20.
There exist absolute constants
C, c > such that for any n ≥ C and any x ∈ (0 , ∞ ] satisfying nx ≥ C the following holds. Let γ ⊂ T \ T be a circuit, let H (cid:48) be a domain and set E := LoopConf ( H (cid:48) , ω ∩ Int E ( γ )) . Then, for any integers k ≥ (cid:96) ≥ , we have P ∅ Int( γ ) ,n,x (cid:0) ∂ Int V ( γ ) ⊂ V ( ω, γ ) , | V ( ω, γ ) | ≥ k and | V (cid:48) ( ω, γ ) | ≥ (cid:96) | E (cid:1) ≤ ( cn · min { x , } ) − k/ max { x (cid:96) , } . Proof.
Let γ ⊂ T \ T be a circuit and denote H := Int( γ ). Let n > x ∈ (0 , ∞ ]. We mayassume throughout the proof that n · min { x , } is sufficiently large, as otherwise the statement istrivial. We shall show that for any ∅ (cid:54) = V ⊂ Int V ( γ ), P ∅ H,n,x ( V ( ω, γ ) = V and | V (cid:48) ( ω, γ ) | ≥ (cid:96) | E ) ≤ (2 √ | V | · ( n · min { x , } ) −| V | / max { x (cid:96) , } . (17) XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n In light of Lemma 2.19 and Lemma 1.10, Lemma 2.20 will then follow from (17) by summing overall sets V with ∂ Int V ( γ ) ⊂ V ⊂ Int V ( γ ) such that V is connected in H × and has cardinality at least k . In order to prove (17), we shall apply Lemma 1.9 to the (restricted) repair map R γ : { ω ∈ LoopConf ( H, ∅ ) ∩ E : V ( ω, γ ) = V and | V (cid:48) ( ω, γ ) | ≥ (cid:96) } → LoopConf ( H, ∅ ) ∩ E , which, by Lemma 2.14, is well-defined. By Proposition 2.15, we may take p := ( n · min { x , } ) | V | / · max { x (cid:96) , } . It remains to estimate, for each V , the maximum number of preimages under R γ of agiven loop configuration.Let ω be such that V ( ω, γ ) = V and let E ( V ) be the set of edges with both endpoints in V .We claim that the set ω \ E ( V ) may be reconstructed from R γ ( ω ) and V . Indeed, ω ⊂ Int E ( γ )since ω ∈ LoopConf ( H, ∅ ), and, for every e ∈ Int E ( γ ) \ E ( V ), we may determine whether e ∈ ω in the following way. Since e has an endpoint u ∈ Int V ( γ ) \ V , we see that e belongs to a c -cluster E of ω for some c ∈ { , , } . In this case, ω ∩ E equals either R γ ( ω ) ∩ E , R γ ( ω ) ↑ ∩ E or R γ ( ω ) ↓ ∩ E , depending on whether c = 0, c = 1 or c = 2, respectively. Hence, it suffices todetermine c from V . To this end, consider a path from u to V in Int( γ ), and let { u, v } be the firstedge on this path such that u / ∈ V and v ∈ V . Observe that u ∈ Int V ( σ ( E )) and v ∈ Ext V ( σ ( E ))since ∂ Ext V ( σ ( E )) ∩ Int V ( γ ) ⊂ V ⊂ Ext V ( σ ( E )) by (4) and the definition of V ( ω, γ ). Thus, { u, v } ∈ σ ( E ) ∗ . Finally, since σ ( E ) ⊂ T \ T c , we see that c is the unique element in { , , } suchthat y, z / ∈ T c , where { y, z } ∗ = { u, v } .In conclusion, since given V ( ω, γ ) = V , R γ ( ω ) uniquely determines ω \ E ( V ), the number ofpreimages of a given loop configuration R γ ( ω ) is at most the number of subsets of E ( V ). Sincethere are at most 3 | V | / V , there are at most 2 | V | / subsets of E ( V ).Thus, Lemma 1.9 implies (17). (cid:3) Proof of Lemma 2.10.
Let A be the event that ∂ Int V ( γ ) ⊂ V ( ω, γ ) and | V ( ω, γ ) | ≥ k . Denote E := LoopConf ( H, ω ∩ Int E ( γ )). Using the fact that γ is vacant in ω , the domain Markovproperty implies that P H,n,x ( A | γ vacant) = P ∅ Int( γ ) ,n,x ( A | E ) . Thus, the result follows from Lemma 2.20. (cid:3) Proofs of main theorems
Throughout this section, we continue to use the notation introduced in Section 2.1. The proofsof the main theorems mostly rely on the main lemma, Lemma 2.10.3.1.
Exponential decay of loop lengths.
As mentioned in the introduction, the results for small x follow via a Peierls argument. The following lemma gives an upper bound on the probability thata given collection of loops appears in a random loop configuration. Lemma 3.1.
Let H be a domain and let ξ be a loop configuration. Then, for any n > , any x > and any A ∈ LoopConf ( H, ∅ ) , we have P ξH,n,x ( A ⊂ ω ) ≤ n L H ( A ) x o H ( A ) . Proof.
Consider the map T : { ω ∈ LoopConf ( H, ξ ) : A ⊂ ω } → LoopConf ( H, ξ )defined by T ( ω ) := ω \ A. Clearly, T is well-defined (see Lemma 2.5e) and injective. Moreover, since L H ( T ( ω )) = L H ( ω ) − L H ( A ) and o H ( T ( ω )) = o H ( ω ) − o H ( A ), we have P ξH,n,x ( T ( ω )) = P ξH,n,x ( ω ) · n − L H ( A ) x − o H ( A ) . Hence, the statement follows from Lemma 1.9. (cid:3)
Recall the notion of a loop surrounding a vertex given prior to Theorem 1.4.
Corollary 3.2.
For any n > , any x > , any domain H , any vertex u ∈ V ( H ) and any positiveinteger k , we have P ∅ H,n,x ( there exists a loop of length k surrounding u ) ≤ kn (2 x ) k . Proof.
Denote by a k the number of simple paths of length k in H starting at a given vertex. Clearly, a k ≤ · k − . It is then easy to see that the number of loops of length k surrounding u is at most ka k − ≤ k k . Thus, the result follows by the union bound and Lemma 3.1. (cid:3) Our main lemma, Lemma 2.10, shows that for a given circuit γ (with a type) it is unlikely thatthe set V ( ω, γ ) is large. The set V ( ω, γ ) specifies deviations from the ground states which are‘visible’ from γ , i.e., deviations which are not ‘hidden’ inside clusters. In Theorem 1.4, we claimthat it is unlikely to see long loops surrounding a given vertex. Any such long loop constitutes adeviation from all ground states. Thus, the theorem would follow from the main lemma (in the maincase, when x is large) if the long loop was captured in V ( ω, γ ). Our next lemma bridges the gapbetween the main lemma and the theorem, by showing that even when a deviation is not capturedby V ( ω, γ ), there is necessarily a smaller circuit σ which captures it in V ( ω, σ ). Lemma 3.3.
Let ω be a loop configuration, let c ∈ { , , } and let γ ⊂ T \ T c be a vacant circuitin ω . Let U ⊂ Int V ( γ ) be non-empty and connected and assume that no vertex in U belongs to atrivial loop in ω . Then there exists c (cid:48) ∈ { , , } and a circuit σ ⊂ T \ T c (cid:48) such that Int( σ ) ⊂ Int( γ ) , σ is vacant in ω and U ∪ ∂ Int V ( σ ) ⊂ V ( ω, σ ) .Proof. We prove the lemma by induction on | Int V ( γ ) | . We consider two cases.Assume first that ∂ Int V ( γ ) ⊂ V ( ω, γ ). If U ⊂ V ( ω, γ ) then we are done, with σ = γ . Otherwise,since U is connected and no vertex in U belongs to a trivial loop in ω it follows that U is disjoint from V ( ω, γ ). Thus, using again the connectedness of U and (4), there is a cluster E of ω inside γ whichcontains all edges incident to vertices in U . Denote γ (cid:48) := σ ( E ) and observe that Int( γ (cid:48) ) (cid:40) Int( γ )and that γ (cid:48) is vacant in ω by Lemma 2.5b. Hence, the lemma follows by applying the inductionhypothesis with γ (cid:48) replacing γ .Assume now that ∂ Int V ( γ ) \ V ( ω, γ ) (cid:54) = ∅ . Let u ∈ ∂ Int V ( γ ) \ V ( ω, γ ) and note that u necessarilyborders a c -flower z of ω . Consider the subgraph H (cid:48) induced by the vertices of H which do notborder z . Observe that U ⊂ V ( H (cid:48) ) and, while H (cid:48) is not necessarily connected, each of its connectedcomponents is a domain of type c . Let γ (cid:48) be the circuit corresponding to the domain containing U .Now Int( γ (cid:48) ) ⊂ H (cid:48) (cid:40) Int( γ ) and γ (cid:48) is vacant in ω as γ is vacant and z is a c -flower. Thus, the lemmafollows by applying the induction hypothesis with γ (cid:48) replacing γ . (cid:3) Proof of Theorem 1.4.
Suppose that n is a sufficiently large constant, let n ≥ n and let x ∈ (0 , ∞ ] be arbitrary. Let c ∈ { , , } , let H be a domain of type c and let u ∈ V ( H ). Weshall estimate the probability that, in a random loop configuration drawn from P ∅ H,n,x , the vertex u is surrounded by a non-trivial loop of length k . We consider two cases, depending on the relativevalues of n and x .Suppose first that nx < n / . Since n ≥ n , we may assume that 2 x ≤ n − / and that kn − k/ ≤ k >
0. By Corollary 3.2, for every k ≥ P ∅ H,n,x (there exists a loop of length k surrounding u ) ≤ kn (2 x ) k ≤ kn − k/ ≤ kn − k/ ≤ n − k/ . XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n We now assume that nx ≥ n / . Since n ≥ n , we may assume that n · min { x , } is sufficientlylarge for our arguments to hold. Let L ⊂ H be a non-trivial loop of length k surrounding u . Notethat, if ω ∈ LoopConf ( H, ∅ ) has L ⊂ ω then, by Lemma 3.3, for some c (cid:48) ∈ { , , } , there exists acircuit σ ⊂ T \ T c (cid:48) such that Int( σ ) ⊂ H , σ is vacant in ω and V ( L ) ∪ ∂ Int V ( σ ) ⊂ V ( ω, σ ). Usingthe fact that H is of type c and the equivalence (3), the domain Markov property and Lemma 2.10imply that for every fixed circuit σ ⊂ T \ T c (cid:48) with Int( σ ) ⊂ H , P ∅ H,n,x ( σ vacant and V ( L ) ∪ ∂ Int V ( σ ) ⊂ V ( ω, σ )) ≤ ( cn · min { x , } ) −| V ( L ) ∪ ∂ Int V ( σ ) | / . Thus, denoting by G ( u ) the set of circuits σ contained in T \ T c (cid:48) for some c (cid:48) ∈ { , , } and having u ∈ Int V ( σ ), we obtain P ∅ H,n,x ( L ⊂ ω ) ≤ (cid:88) σ ∈G ( u ) ( cn · min { x , } ) −| V ( L ) ∪ ∂ Int V ( σ ) | / ≤ ∞ (cid:88) (cid:96) =1 D (cid:96) ( cn · min { x , } ) − max { (cid:96),k } / ≤ ( c (cid:48) n · min { x , } ) − k/ , where we used the facts that the length of a circuit σ such that | ∂ Int V ( σ ) | = (cid:96) is at most 3 (cid:96) , that thenumber of circuits σ of length at most 3 (cid:96) with u ∈ Int V ( σ ) is bounded by D (cid:96) for some sufficientlylarge constant D , and in the last inequality we used the assumption that n · min { x , } is sufficientlylarge. Since the number of loops of length k surrounding a given vertex is smaller than k k (see theproof of Corollary 3.2), our assumptions that nx ≥ n / and n ≥ n yield P ∅ H,n,x (there exists a loop of length k surrounding u ) ≤ k k ( c (cid:48) n / ) − k/ ≤ n − k/ . Proof of Theorem 1.5.
The proof is very similar to that of Theorem 1.4. The main difference isthe following replacement of Lemma 2.10. Recall that in every λ ∈ LoopConf ( H, ∅ , u, v ), there is asimple path between u and v . Let p ( λ, u, v ) be such a path and denote ω λ := λ \ E ( p ( λ, u, v )), sothat ω λ ∈ LoopConf ( H, ∅ ) and L (cid:48) H ( λ ) = L H ( ω λ ). For a circuit γ for which Int( γ ) ⊂ H and for apositive integer k , let E ( H, u, v, γ, k ) be the set of configurations λ ∈ LoopConf ( H, ∅ , u, v ) such that • γ is vacant in ω λ ; • V ( p ( λ, u, v )) \ { u, v } and ∂ Int V ( γ ) are contained in V ( ω λ , γ ); • | V ( ω λ , γ ) | ≥ k .For ω ∈ LoopConf ( H, ∅ ) and λ ∈ LoopConf ( H, ∅ , u, v ), denote φ H,n,x ( ω ) := x o H ( ω ) n L H ( ω ) ,φ H,n,x ( λ ) := x o H ( λ ) n L (cid:48) H ( λ ) J ( λ ) . Lemma 3.4.
There exist absolute constants
C, c > such that for any n ≥ C and x ∈ (0 , ∞ ) satisfying nx ≥ C the following holds. For any domain H , any c ∈ { , , } , any circuit γ ⊂ T \ T c for which Int( γ ) ⊂ H , any distinct vertices u, v ∈ V ( H ) and any positive integer k , we have (cid:88) λ ∈E ( H,u,v,γ,k ) φ H,n,x ( λ ) ≤ x ( cn · min { x , } ) − k/ (cid:88) ω ∈ LoopConf ( H, ∅ ) φ H,n,x ( ω ) . Proof.
By symmetry, it suffices to consider the case that c = 0. For (cid:96) ≥
0, let E (cid:96) denote theset of λ ∈ E ( H, u, v, γ, k ) having | V ( p ( λ, u, v )) \ { u, v }| = (cid:96) and set E (cid:48) (cid:96) := { ω λ : λ ∈ E (cid:96) } . Since V ( p ( λ, u, v )) \ { u, v } ⊂ V ( ω λ , γ ), we have | V (cid:48) ( ω λ , γ ) | ≥ (cid:96) for any λ ∈ E (cid:96) . Therefore, by Lemma 2.20, (cid:88) ω ∈E (cid:48) (cid:96) φ H,n,x ( ω ) ≤ ( cn · min { x , } ) − k/ · max { x, } − (cid:96) · (cid:88) ω ∈ LoopConf ( H, ∅ ) φ H,n,x ( ω ) . Since J ( λ ) ≤ | E ( p ( λ, u, v )) | = (cid:96) + 1 for any λ ∈ E (cid:96) , we have φ H,n,x ( λ ) = φ H,n,x ( ω λ ) · x | E ( p ( λ,u,v )) | J ( λ ) ≤ φ H,n,x ( ω λ ) · x · max { x, } (cid:96) . Thus, noting that for every ω ∈ E (cid:48) (cid:96) , |{ λ ∈ E (cid:96) : ω λ = ω }| ≤ (cid:96) + 1 from u to v ) ≤ · (cid:96) − ≤ (cid:96) +1 , we obtain (cid:88) λ ∈E (cid:96) φ H,n,x ( λ ) ≤ x · (cid:96) +1 · ( cn · min { x , } ) − k/ · (cid:88) ω ∈ LoopConf ( H, ∅ ) φ H,n,x ( ω ) . Finally, the lemma follows by summing over 0 ≤ (cid:96) ≤ k . (cid:3) We shall also require the following replacement of Corollary 3.2.
Lemma 3.5.
Let n > and < x ≤ . For any domain H and any distinct u, v ∈ V ( H ) , we have (cid:88) λ ∈ LoopConf ( H, ∅ ,u,v ) φ H,n,x ( λ ) ≤ x ) d H ( u,v ) (cid:88) ω ∈ LoopConf ( H, ∅ ) φ H,n,x ( ω ) . Proof.
The number of possibilities for a simple path of length k from u to v is at most 3 · k − .Consideration of the map λ (cid:55)→ ω λ , the fact that J ( λ ) ≤ p ( λ, u, v ) now shows that the ratio of the sums appearing in the lemma is bounded above by (cid:88) k ≥ d H ( u,v ) x k (3 · k − ) = 94 · (2 x ) d H ( u,v ) − x ≤ x ) d H ( u,v ) . (cid:3) We now proceed along the same lines as the proof of Theorem 1.4. Suppose first that nx < n / .Since n ≥ n , the theorem follows as an immediate consequence of Lemma 3.5. Suppose nowthat nx ≥ n / . For each λ ∈ LoopConf ( H, ∅ , u, v ), by Lemma 3.3 applied to ω λ , there exists acircuit σ ⊂ T \ T c (cid:48) for some c (cid:48) ∈ { , , } such that Int( σ ) ⊂ H and λ ∈ E ( H, u, v, σ, k σ ), where k σ := max { d H ( u, v ) − , | ∂ Int V ( σ ) |} . The theorem now follows with a similar calculation as inTheorem 1.4, by summing over all possibilities for the circuit σ and applying Lemma 3.4 with γ = σ and k = k σ .3.2. Small perturbation of ground state.Proof of Theorem 1.8.
By definition, the subgraph of H induced by C ( ω, u ) is a domain when itis non-empty. Let Γ( ω, u ) be the circuit satisfying C ( ω, u ) = Int V (Γ( ω, u )). It follows that Γ( ω, u )is vacant and contained in T \ T . To see this, note that the edge boundary of B ( ω ) consists only ofedges { v, w } such that w borders a 0-flower y and v is the unique neighbor of v not bordering y ; inparticular, { v, w } borders a hexagon from T and a hexagon from T and { v, w } (cid:54)∈ ω . Furthermore, ∂ C ( ω, u ) ⊂ V ( ω, Γ( ω, u )). This follows as Γ( ω, u ) is vacant in ω and, by the definition of B ( ω ), novertex of ∂ Int V (Γ( ω, u )) belongs to a trivial loop surrounding a hexagon in T . XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n Now, denoting by G k ( u ) the set of circuits γ ⊂ T \ T having u ∈ Int V ( γ ) and | ∂ Int V ( γ ) | ≥ k ,Lemma 2.10 implies that P H,n,x ( | ∂ C ( ω, u ) | ≥ k ) = (cid:88) γ ∈G k ( u ) P H,n,x (Γ( ω, u ) = γ ) ≤ (cid:88) γ ∈G k ( u ) P H,n,x ( γ vacant and ∂ Int V ( γ ) ⊂ V ( ω, γ )) ≤ (cid:88) γ ∈G k ( u ) ( cn · min { x , } ) −| ∂ Int V ( γ ) | / ≤ (cid:88) (cid:96) ≥ k D (cid:96) ( cn · min { x , } ) − (cid:96)/ ≤ ( c (cid:48) n · min { x , } ) − k/ , where c (cid:48) , D are positive constants. In the final inequality, we used the facts that the length of acircuit γ such that | ∂ Int V ( γ ) | = (cid:96) is at most 3 (cid:96) , and that the number of circuits of length at most3 (cid:96) surrounding u is bounded by D (cid:96) for some sufficiently large constant D .3.3. Limiting Gibbs measures.
Before proving the last two theorems, we require the followingtwo lemmas. We say that a circuit γ surrounds a subgraph A ⊂ H if A ⊂ Int( γ ) and that γ is inside A if Int( γ ) ⊂ A . We say that a circuit γ contains a circuit σ if Int( σ ) ⊂ Int( γ ). Lemma 3.6.
Let H and H (cid:48) be two domains, let A ⊂ H ∩ H (cid:48) be a non-empty subgraph and let ξ and ξ (cid:48) be loop configurations. Let n > and x ∈ (0 , ∞ ] . Let ω ∼ P ξH,n,x and ω (cid:48) ∼ P ξ (cid:48) H (cid:48) ,n,x be independent.Denote by Ω the event that there exists a circuit surrounding A and inside H ∩ H (cid:48) which is vacantin both ω and ω (cid:48) . Assume that Ω has positive probability. Then, conditioned on Ω , the marginaldistributions of ω and ω (cid:48) on A are equal.Proof. In this proof, a doubly-vacant circuit is a circuit which is vacant in both ω and ω (cid:48) . Let G denote the collection of circuits surrounding A and inside H ∩ H (cid:48) . Let σ, σ (cid:48) ∈ G be doubly-vacantcircuits. Then, since both circuits surround A , Int( σ ) ∩ Int( σ (cid:48) ) (cid:54) = ∅ . By Fact 2.3, there exists acircuit γ having γ ∗ ⊂ σ ∗ ∪ ( σ (cid:48) ) ∗ which contains both σ and σ (cid:48) . Clearly, γ is doubly-vacant, surrounds A and is inside H ∩ H (cid:48) , and hence γ ∈ G . Thus, we have a notion of the “outermost” doubly-vacantcircuit in G . On Ω, define Γ to be this circuit. Then, we claim that, for any circuit γ ∈ G for whichthe event Ω ∩{ Γ = γ } has positive probability, conditioned on Ω ∩{ Γ = γ } , the marginal distributionof ( ω, ω (cid:48) ) on A is the same as the marginal distribution of two independent loop configurationssampled from P ∅ Int( γ ) ,n,x . Indeed, since the event Ω ∩ { Γ = γ } is determined by ω \ Int E ( γ ) and ω (cid:48) \ Int E ( γ ), this follows from the domain Markov property. (cid:3) Lemma 3.7.
Let H k be an increasing sequence of domains such that ∪ k H k = H and let ξ k bea sequence of loop configurations. Let n > and x ∈ (0 , ∞ ] and assume that P ξ k H k ,n,x converges(weakly) as k → ∞ to an infinite-volume measure P which is supported on loop configurations withno infinite paths. Then P is a Gibbs measure for the loop O ( n ) model with edge weight x .Proof. For a domain H , denote by F H the sigma algebra generated by the events { e ∈ ω } for e ∈ E ( H ). For a loop configuration τ , let E τm be the event that ω and τ coincide on E ( H m ) \ E ( H ).By L´evy’s zero-one law, P is a Gibbs measure if and only if for every domain H and every A ∈ F H ,lim m →∞ P ( A | E τm ) = P τH,n,x ( A ) for P -almost every τ . Fix a domain H and A ∈ F H . By the definition of P , we need to show thatlim m →∞ lim k →∞ P ξ k H k ,n,x ( A | E τm ) = P τH,n,x ( A ) for P -almost every τ . Indeed, for any τ having a vacant circuit γ with H ⊂ Int( γ ), the domain Markov property impliesthat P ξ k H k ,n,x ( A | E τm ) = P τH,n,x ( A ) for large enough m and k ≥ m . As P is supported on loop configurations with no infinite paths, such a circuit exists for P -almost every τ (consider the smallestdomain containing V ( H ) and all the connected components of τ which intersect V ( H ) and applyFact 2.2). (cid:3) Proof of Theorem 1.6.
We start with a lemma.
Lemma 3.8.
Let n > and x > . For any two domains H and H (cid:48) , any vertex u ∈ V ( H ) and anypositive integer k , we have P ( the connected component of u in ω ∪ ω (cid:48) has exactly k edges ) ≤ (9 e max { n / , } x ) k , where ω ∼ P ∅ H,n,x and ω (cid:48) ∼ P ∅ H (cid:48) ,n,x are independent.Proof. We may assume that max { n / , } x ≤
1, since the statement is trivial otherwise. Let C k bethe set of connected subgraphs of H that have exactly k edges and contain u . For S ∈ C k , call a pairof loop configurations ( A, A (cid:48) ) compatible with S if E ( A ) ∪ E ( A (cid:48) ) = E ( S ). Let S be the connectedcomponent of u in ω ∪ ω (cid:48) . Then P ( | E ( S ) | = k ) ≤ (cid:88) S ∈C k (cid:88) ( A,A (cid:48) ) compatible with S P ( A ⊂ ω, A (cid:48) ⊂ ω (cid:48) ) ≤ (cid:88) S ∈C k (cid:88) ( A,A (cid:48) ) compatible with S (max { n / , } x ) o H ( A )+ o H (cid:48) ( A (cid:48) ) ≤ (9 e ) k (max { n / , } x ) k . The second inequality follows from Lemma 3.1 and the facts that ω and ω (cid:48) are independent and thatany loop consists of at least six edges. The last inequality follows from the following three facts: • o H ( A ) + o H (cid:48) ( A (cid:48) ) ≥ | E ( S ) | = k and max { n / , } x ≤ • the number of possible pairs of loop configurations ( A, A (cid:48) ) compatible with S is bounded by3 k (since each edge in S must be in either A , A (cid:48) or in both); • |C k | is bounded by 3(3 e ) k − ≤ (3 e ) k (apply Lemma 1.10 to the 4-regular line graph of H ,using an edge incident to u as the given vertex). (cid:3) Let us conclude the proof of Theorem 1.6. Assume that 9 e max { n / , } x ≤ /e . Let H and H (cid:48) be two domains and let A ⊂ B ⊂ H ∩ H (cid:48) be two sub-domains. Let ω ∼ P ∅ H,n,x and ω (cid:48) ∼ P ∅ H (cid:48) ,n,x beindependent. Let E be the event that the union of the connected components of the vertices of A in the graph ω ∪ ω (cid:48) intersects V ( H ) \ V ( B ). Lemma 3.8 implies that P ( E ) ≤ (cid:88) v ∈ V ( A ) ∞ (cid:88) k = d ( { v } ,V ( H ) \ V ( B )) (9 e max { n / , } x ) k ≤ | V ( A ) | · e − d ( A,V ( H ) \ V ( B )) , (18)where d ( E, F ) is the minimum of the graph distances between a vertex in E and a vertex in F .Let us now show that, on the complement of E , there exists a circuit γ surrounding A and inside H ∩ H (cid:48) which is vacant in both ω and ω (cid:48) . We first define the notion of the outer circuit of a non-emptyfinite connected subset U of V ( H ). Let U (cid:48) be the unique infinite connected component of V ( H ) \ U and let U (cid:48)(cid:48) := V ( H ) \ U (cid:48) . Evidently, the subgraph of H induced by U (cid:48)(cid:48) is a domain containing U .The outer circuit σ of U is then the circuit corresponding to this domain, i.e., U (cid:48)(cid:48) = Int V ( σ ), whichexists by Fact 2.2. Note also that ∂U (cid:48)(cid:48) ⊂ ∂U and that if U is contained in some domain then U (cid:48)(cid:48) isalso contained in the same domain.Let D be the union of the connected components of vertices of A in ω ∪ ω (cid:48) . Let γ be the outercircuit of V ( A ) ∪ D , and note that, on the complement of E , γ is inside B . Let us show that γ is vacant in both ω and ω (cid:48) . To this end, let e = ( u, v ) ∈ γ ∗ be an edge with u ∈ V ( A ) ∪ D and v / ∈ V ( A ) ∪ D . Assume first that u ∈ D . Clearly e / ∈ ω ∪ ω (cid:48) , as otherwise, v would also belong to D .Assume now that u ∈ V ( A ) \ D . Then, by definition of D , u is not contained in a loop of neither ω nor ω (cid:48) . In particular, e does not belong to neither ω nor ω (cid:48) . Thus, γ is vacant in both ω and ω (cid:48) . XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n Thus, by Lemma 3.6, the total variation between the measures P ∅ H,n,x ( · | A ) and P ∅ H (cid:48) ,n,x ( · | A ) is atmost P ( E ). In light of (18), by taking B large enough, we may make P ( E ) arbitrarily small. Thisimplies the convergence of the measures P ∅ H k ,n,x ( · | A ) towards a limit. Since this holds for any domain A , we have established the convergence of P ∅ H k ,n,x as k → ∞ towards an infinite-volume measure P H ,n,x .The fact that P H ,n,x is supported on loop configurations with no infinite paths is an immediateconsequence of Corollary 3.2. Indeed, the corollary shows that in the measure P ∅ H k ,n,x , the probabilitythat a given vertex is contained in a loop of length m tends to zero with m , uniformly in k . Finally,the fact that P H ,n,x is a Gibbs measure follows from Lemma 3.7. Proof of Theorem 1.7.
Let us first assume that the convergence to the limiting measures { P c H ,n,x } c ∈{ , , } holds and deduce the properties of these measures when n · min { x , } is sufficiently large. By The-orem 1.8, if n · min { x , } is sufficiently large then, for any z ∈ T , P H ,n,x ( z is surrounded by a trivial loop) > / . Since P H ,n,x and P H ,n,x are the measures induced by applying the shifts ↓ and ↑ , respectively, to P H ,n,x , the same statement holds for any P c H ,n,x with z ∈ T c . Thus, since adjacent hexagons cannotboth be surrounded by trivial loops simultaneously, we conclude that the measures { P c H ,n,x } c ∈{ , , } are not convex combinations of one another. Next, the fact that P c H ,n,x is supported on loop con-figurations with no infinite paths is an immediate consequence of Theorem 1.4 (by using (3) andapplying the convergence result with an exhausting sequence of domains of type c ). Finally, the factthat P c H ,n,x is a Gibbs measure follows from Lemma 3.7.It remains to show that, for any c ∈ { , , } , P c H k ,n,x converges as k → ∞ to an infinite-volumemeasure P c H ,n,x . Without loss of generality, we may assume that c = 0. The proof bears similaritywith the proof of Theorem 1.6.We start with a lemma. Recall the definition of B ( ω ) and C ( ω, u ) from Section 1.1 and recall thedefinition of H × from Section 2.5. For a domain H and a loop configuration ω ∈ LoopConf ( H, ω ),set C ( ω ) := V ( H ) \ B ( ω ) = ∪ u ∈ V ( H ) C ( ω, u ). Note that, by definition, every two breakups C ( ω, u ) and C ( ω, v ), where u, v ∈ V ( H ), are either equal or their union is disconnected in H × (as the definitionimplies that if a vertex belongs to C ( ω ) then all vertices bordering the same hexagon in T alsobelong to C ( ω )). Thus, every connected component of C ( ω ) is a breakup of some vertex, and every H × -connected component of ∂ C ( ω ) is the boundary of a breakup of some vertex, i.e., equals ∂ C ( ω, u )for some u ∈ V ( H ) (recall that this set is H × -connected, by Lemma 2.18). Lemma 3.9.
There exists an absolute constant c > such that for any n > and x ∈ (0 , ∞ ] thefollowing holds. For any two domains H and H (cid:48) , any vertex u ∈ V ( H ) and any positive integer k , P ( the H × -connected component of u in ∂ C ( ω ) ∪ ∂ C ( ω (cid:48) ) has cardinality k ) ≤ ( cn · min { x , } ) − k/ , where ω ∼ P H,n,x and ω (cid:48) ∼ P H (cid:48) ,n,x are independent.Proof. Let C k be the set of H × -connected subsets of V ( H ) of cardinality k containing u . For S ∈ C k ,call a pair ( A, A (cid:48) ) of subsets of V ( H ) compatible with S if A ∪ A (cid:48) = S . We write A ≺ C ( ω ) if A is the union of some H × -connected components of ∂ C ( ω ), or equivalently, if every H × -connectedcomponent of A is equal to ∂ C ( ω, v ) for some v ∈ V ( H ). Now, we claim that for each fixed A , wehave P H,n,x ( A ≺ C ( ω )) ≤ ( cn · min { x , } ) −| A | / . (19)To see this, note that for the probability to be positive, A needs to be a union of ∂ Int V ( γ i ) for acollection of circuits γ i ⊂ T \ T with disjoint interiors. Moreover, on the event A ≺ C ( ω ) thesecircuits are necessarily vacant in ω . Therefore, by conditioning on all of the γ i being vacant, we may apply the domain Markov property and Theorem 1.8 to obtain the estimate (19). Similarly,for each fixed A (cid:48) we have that P H (cid:48) ,n,x ( A (cid:48) ≺ C ( ω (cid:48) )) ≤ ( cn · min { x , } ) −| A (cid:48) | / . We may assume that cn · min { x , } ≥
1, since the statement is trivial otherwise. Let S be the H × -connected component of u in ∂ C ( ω ) ∪ ∂ C ( ω (cid:48) ). Then P ( | S | = k ) ≤ (cid:88) S ∈C k (cid:88) ( A,A (cid:48) ) compatible with S P ( A ≺ C ( ω ) , A (cid:48) ≺ C ( ω (cid:48) )) ≤ (cid:88) S ∈C k (cid:88) ( A,A (cid:48) ) compatible with S ( cn · min { x , } ) − ( | A | + | A (cid:48) | ) / ≤ (15 e ) k ( cn · min { x , } ) − k/ . In the second inequality we used the fact that ω and ω (cid:48) are independent. The last inequality followsfrom the following three facts: • | A | + | A (cid:48) | ≥ | S | = k and cn · min { x , } ≥ • the number of possible pairs ( A, A (cid:48) ) compatible with S is bounded by 3 k (since each vertexin S is either in A , in A (cid:48) or in both); • |C k | is bounded by (5 e ) k − ≤ (5 e ) k (apply Lemma 1.10 to the 6-regular graph H × ). (cid:3) Let us conclude the proof of Theorem 1.7. Let c > cn · min { x , } ≥ e . Let H and H (cid:48) be two domains and let A ⊂ B ⊂ H ∩ H (cid:48) be two domains of type 0. Let ω ∼ P H,n,x and ω (cid:48) ∼ P H (cid:48) ,n,x be independent. Let E be the event that the union of H × -connected components ofvertices in A in ∂ C ( ω ) ∪ ∂ C ( ω (cid:48) ) intersects V ( H ) \ V ( B ). Lemma 3.9 implies that P ( E ) ≤ (cid:88) u ∈ V ( A ) ∞ (cid:88) k = d ( { u } ,V ( H ) \ V ( B )) ( cn · min { x , } ) − k/ ≤ | V ( A ) | · e − d ( A,V ( H ) \ V ( B )) , where d ( E, F ) is the minimum of the graph distances between a vertex in E and a vertex in F .Let E (cid:48) be the event that A is contained in either C ( ω ) or C ( ω (cid:48) ), i.e., that A is contained entirely inone breakup (of either ω or ω (cid:48) ). Denote by ρ ( m ) the smallest possible size of ∂U for a finite subset U ⊂ V ( H ) of size at least m . Then Theorem 1.8 implies that P ( E (cid:48) ) ≤ cn · min { x , } ) − ρ ( | V ( A ) | ) / ≤ e − ρ ( | V ( A ) | ) . Let us now show that, on the complement of
E ∪ E (cid:48) , there exists a circuit γ ⊂ T \ T surrounding A and inside H ∩ H (cid:48) which is vacant in both ω and ω (cid:48) . We require the following simple geometricclaim. For brevity, in the rest of the proof we identify a domain with its set of vertices.If S, T are two domains of type 0 with S (cid:54)⊂ T and T (cid:54)⊂ S such that S ∪ T is connected,then ∂S ∪ ∂T is H × -connected. If, in addition, S ∩ T (cid:54) = ∅ then also ∂S ∩ T (cid:54) = ∅ . (20)To see this, note first that ∂S and ∂T are H × -connected by Fact 2.2 and Lemma 2.18. If S ∩ T = ∅ then the assumption that S ∪ T is connected implies that a vertex of ∂S is adjacent to a vertex of ∂T yielding that ∂S ∪ ∂T is H × -connected. Assume that S ∩ T (cid:54) = ∅ . By considering a path in T from T \ S to T ∩ S it follows that ∂S ∩ T (cid:54) = ∅ . Similarly, considering a path in T c from S \ T to( S ∪ T ) c shows that ∂S \ T (cid:54) = ∅ . Finally, by considering a H × -path in ∂S from ∂S ∩ T to ∂S \ T ,we see that either ∂S ∩ ∂T (cid:54) = ∅ or a vertex of ∂S is adjacent to a vertex of ∂T . In either case, weconclude that ∂S ∪ ∂T is H × -connected.Recall the notion of the outer circuit of a non-empty finite connected subset U of V ( H ) from theproof of Theorem 1.6. Let D be the union of A and of the connected components of C ( ω ) ∪ C ( ω (cid:48) )that intersect A . Let γ be the outer circuit of D . It follows that γ ⊂ T \ T and that γ is vacantin both ω and ω (cid:48) . Indeed, γ ⊂ T \ T since A is a domain of type 0 and, by the definition of the XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n breakup, each of the C ( ω, u ) is a domain of type 0. Thus, no edge of γ ∗ can belong to ω ∪ ω (cid:48) sinceotherwise both its endpoints would belong to a breakup.We claim that, on the complement of E ∪ E (cid:48) , γ is inside B . By the definition of γ and since B is adomain, it suffices to show that D ⊂ B . On the complement of E (cid:48) , we may write D as the union ofdomains D i of type 0 such that no one contains another, D = A and each D i , i (cid:54) = 0, is a breakupof either ω or ω (cid:48) . Let D (cid:48) be the union of A and of the H × -connected components of ∂ C ( ω ) ∪ ∂ C ( ω (cid:48) )that intersect A . On the complement of E , we have D (cid:48) ⊂ B . By (20), ∪ i ∂D i is H × -connected andif D i ∩ A (cid:54) = ∅ then ∂D i ∩ A (cid:54) = ∅ . Thus ∪ i ∂D i ⊂ D (cid:48) . We conclude that ∂ D ⊂ ∪ i ∂D i ⊂ B , whence D ⊂ B as we wanted to show.Thus, by Lemma 3.6, the total variation between the measures P H,n,x ( · | A ) and P H (cid:48) ,n,x ( · | A ) is atmost P ( E ∪ E (cid:48) ). In particular, fixing a subgraph A (cid:48) ⊂ A , the same holds for the measures P H,n,x ( · | A (cid:48) )and P H (cid:48) ,n,x ( · | A (cid:48) ). Since ρ ( m ) clearly tends to infinity as m tends to infinity, by first taking A largeenough and then taking B large enough, we may make P ( E ∪ E (cid:48) ) arbitrarily small. This implies theconvergence of the measures P H k ,n,x ( · | A (cid:48) ) towards a limit. Since this holds for any finite subgraph A (cid:48) of H , we have established the convergence of P H k ,n,x as k → ∞ towards an infinite-volume measure P H ,n,x . 4. Discussion and open questions
In this work, we investigate the structure of loop configurations in the loop O ( n ) model withlarge parameter n . We show that the chance of having a loop of length k surrounding a given vertexdecays exponentially in k . In addition, we show, under appropriate boundary conditions, that if nx is small, the model is in a dilute, disordered phase whereas if nx is large, configurations typicallyresemble one of the three ground states. In this section, we briefly discuss several future researchdirections. Spin O(n).
As described in the introduction, the loop O ( n ) model can be viewed as an approxima-tion of the spin O ( n ) model, with the length of loops related to the spin-spin correlation function.Thus, our results prove an analogue of the well-known conjecture that spin-spin correlations decayexponentially (in the distance between the sites) in the planar spin O ( n ) model with n ≥
3, at anypositive temperature. Proving the conjecture itself remains a tantalizing challenge.
Small n . Studying the loop O ( n ) model for small values of n is of great interest. It is predicted thatthe model displays critical behavior only when n ≤
2. There, it is expected to undergo a Kosterlitz–Thouless phase transition at x c = 1 / (cid:112) √ − n , see [29], and exhibit conformal invariance when x ≥ x c . Mathematical results on this are currently restricted to the cases n = 1 and n = 0, whichcorrespond to the Ising model and the self-avoiding walk , respectively. For these two cases, thecritical values have been identified rigorously in [21] and [14], respectively. In the n = 1 case, themodel has been proved [6, 7] to be conformally invariant at x c = 1 / √
3. For n = 1 and x = ∞ theheight function of the model may be viewed as a uniformly chosen lozenge tiling of a domain in theplane. This viewpoint leads to a determinantal process, the dimer model , which has been analyzedin great detail (see, e.g., [19] for an introduction). Conformal invariance has also been proved forthe double dimer model which is closely related to the case n = 2 and x = ∞ (see [20]).Our results are limited to the case n ≥ n and understanding the various behaviors for smallvalues of n remains a beautiful mathematical challenge. To give a taste of the different possibilities,we provide some simulation results in Figure . Extremality and uniqueness of the Gibbs measures.
When n ≥ n and nx ≥ C , we provethat the model has at least three different Gibbs measures, distinguished by a choice of a sublatticeof the triangular lattice. Are these the only extremal Gibbs measures in this regime (i.e., is everyother measure a convex combination of these three measures)? Such a result would be in the spiritof the Aizenman–Higuchi theorem [1, 18] which proves that the only extremal Gibbs measures for (a) n = 0 . x = 0 . (b) n = 0 . x = 0 . (c) n = 2 and x = 1 / √ ≈ . (d) n = 8 and x = 1. Figure 8.
A few samples of random loop configurations. Configurations are on a60 ×
45 domain of type 0 and are sampled via Glauber dynamics for 100 millioniterations started from the empty configuration. The conjectured phase transitionpoint for n = 0 . x c = 1 / (cid:112) √ − . ≈ .
568 and for n = 2 is x c = 1 / √ ≈ . n .the 2D Ising model are the two pure states. This theorem was recently extended to the q -state Pottsmodel in [8].For small values of max { n, } x , we prove the existence of a limiting Gibbs measure when ex-hausting space via an increasing sequence of domains with vacant boundary conditions. Is this Gibbsmeasure unique for each choice of n and x in this regime? Intuitively, the difficulty in proving thislies in dealing with domains with boundary conditions which force an interface (i.e., part of a loop)through the domain (similarly to the situation in Figure ). If this interface passes near the origin XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n with non-negligible probability, one would obtain a limiting Gibbs measure having an infinite pathwith positive probability. However, one expects interfaces to follow diffusive scaling, similarly torandom walk paths, and as such should have negligible probability to pass close to the origin whenthe domain is large. Making such an intuition rigorous is quite non-trivial and was recently carriedout successfully in [8] for planar Potts models. Adapting the ideas in [8] to the loop O ( n ) modelposes a challenge as these rely on specific properties of the Potts model. Roughly, the strategy in [8]proceeds by showing that when starting from a large domain H with arbitrary boundary conditions,only a uniformly bounded number of interfaces will reach the boundary of a smaller sub-domain H (cid:48) . Then it is shown that these bounded number of interfaces follow diffusive scaling as in theintuition above. The first part, bounding the number of interfaces between the boundary of H and H (cid:48) , may possibly be carried out for the loop O ( n ) model by using Lemma 1.9; configurations withmany long interfaces may be ‘rewired’, erasing most of these interfaces and replacing them withshort connections along the boundary of H , yielding configurations with much higher probability.The second part, however, showing the diffusive scaling, remains a major obstacle. The hard-hexagon model.
Our results shed light on the Gibbs measures of the loop O ( n ) modelwhen n ≥ n and either nx ≤ c or nx ≥ C . The structure for n ≥ n and c ≤ nx ≤ C remainsunclear; see Figure and Figure . Is there a single x c ( n ) at which the model transitions from thedilute, disordered phase to the dense, ordered phase? What happens when x = x c ( n )?An intuition for this question may be obtained by considering a limiting model as n tends toinfinity. As noted already in the paper [9] where the loop O ( n ) model was introduced, taking thelimit n → ∞ and nx → λ leads formally to the hard-hexagon model. As loops of length longerthan 6 become less and less likely in this limit, hard-hexagon configurations consist solely of trivialloops, with each such loop contributing a factor of λ to the weight. Thus, the hard-hexagon modelis the hard-core lattice gas model on the triangular lattice T with fugacity λ . For this model, Baxter[2] (see also [3, Chapter 14]) computed the critical fugacity λ c = (cid:16) (cid:16) π (cid:17)(cid:17) = 12 (cid:16)
11 + 5 √ (cid:17) ≈ . , and showed that as λ increases beyond the threshold λ c , the model undergoes a fluid-solid phasetransition from a homogeneous phase in which the sublattice occupation frequencies are equal toa phase in which one of the three sublattices is favored. Additional information is obtained onthe critical behavior including the fact that the mean density of hexagons is equal for each of thethree sublattices [2, Equation (13)] and the fact that the transition is of second order [2, Equation(9)]. Baxter’s arguments use certain assumptions on the model which appear not to have beenmathematically justified. Still, this exact solution may suggest that the loop O ( n ) model with large n will also have a unique transition point x c ( n ), that nx c ( n ) will converge to λ c as n tends toinfinity and that the transition in x is of second order, with the model having a unique Gibbs statewhen x = x c ( n ). Square-lattice random-cluster model and dilute Potts model.
We start with a somewhatinformal description of the square-lattice random-cluster model and refer the interested reader to[16, 12] for more details. The random-cluster model with parameters 0 < p < , q > Z is a random collection of edges η of the domain whose probability is proportional to p o ( η ) (1 − p ) c ( η ) q k ( η ) , where o ( η ) is the number of edges in η , c ( η ) is the number of edges of the domain which are notin η and k ( η ) is the number of connected components in the graph whose vertices are the verticesof the domain and whose edges are given by η . For each η , one may draw a loop configuration ω η (on the so-called medial lattice) consisting of the loops marking the boundaries of the connectedcomponents (these loops go around the connected components and on the boundary of each “hole” Figure 9.
An illustration of a random-cluster configuration η and its correspondingloop configuration ω η . The edges of η are denoted by bold lines, the edges not in η by dashed lines and the loops of ω η by plain lines.that the components surround); see Figure . It turns out that the probability of η may be rewrittenusing these loops so that the probability of η is proportional to λ o ( η ) ( √ q ) L ( ω η ) , (21)where λ := p √ q (1 − p ) and L ( ω η ) is the number of loops in ω η . This representation highlights a selfduality occurring when p is such that λ = 1 and this self-dual point has been proven to be the criticalpoint p c ( q ) for the random-cluster model [4]. The formula (21) may immediately remind the readerof the formula for the probability of configurations in the loop O ( n ) model given in Definition 1.1.However, we emphasize that o ( η ) counts the number of edges in η and as such is quite differentfrom the ‘length’ of the loops in ω η . In fact, the loop configuration ω η is necessarily fully packedin the domain for any given η , so that λ plays a different role from the parameter x of the loop O ( n ) model. Still, the formula (21) does suggest an analogy between the random-cluster model atcriticality (when p = p c ( q )) and the fully packed (i.e., x = ∞ ) loop O ( n ) model with n = √ q .Taken with periodic boundary conditions on a square domain, the random-cluster model has twoconfigurations η which maximize L ( ω η ): one in which all the edges of the domain are absent (yieldingloops around the vertices) and one in which all of them are present (yielding loops around the faces).These configurations are equally probable at the critical point, but one is preferred over the otherwhenever p (cid:54) = p c ( q ). Following a proof of Koteck´y and Shlosman [23] for the closely-related Pottsmodel, it has been proven by Laanait et al. [25] that for large q , the random-cluster model exhibitsa first-order phase transition, so that at criticality there are two Gibbs states corresponding to thetwo ground states described above. Our results on the existence of the ordered phase for large n and x = ∞ are quite analogous to this phenomenon. In fact, it is predicted that the square-latticerandom-cluster model has a first-order phase transition if q ≥ XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n phase transition. This is in line with the conjectured phase diagram for the loop O ( n ) model,predicting that the ordered phase at x = ∞ exists only for n ≥ p of the random-cluster model has no analogue in theloop O ( n ) model and so the existence of a first-order transition in p does not suggest that such atransition should occur also when varying x . As mentioned above, it may well be that for large n ,the transition in x is of second order by analogy with the situation for hard hexagons.Lastly, we mention that Nienhuis [30] proposed a version of the Potts model, termed the dilutePotts model , with a direct relationship to the loop O ( n ) model. A configuration of the dilute Pottsmodel in a domain of the triangular lattice is an assignment of a pair ( s z , t z ) to each vertex z of thedomain, where s z ∈ { , . . . , q } represents a spin and t z ∈ { , } denotes an occupancy variable. Theprobability of configurations involves a hard-core constraint that nearest-neighbor occupied sitesmust have equal spins (reminiscent of the Edwards-Sokal coupling of the Potts and random-clustermodels) and single-site, nearest-neighbor and triangle interaction terms involving the occupancyvariables. With a certain choice of coupling constants, the marginal of the model on the occupancyvariables is equivalent to the loop O ( n ) model (with n = √ q ), with the loops being the interfacesbetween occupied and unoccupied sites. Nienhuis predicts this choice of parameters to be part ofthe critical surface of the dilute Potts model. The properties of the dilute Potts model appear notto have been studied in the mathematical literature and it would be interesting to see whether theycan shed further light on the behavior of the loop O ( n ) model. Height representation for integer n . When the loop parameter n is an integer, the loop O ( n )model admits a height function representation [9]. Let T n be the n -regular tree (so that T = { + , −} and T = Z ) rooted at an arbitrary vertex ρ . Let Lip n be the set of functions ϕ : T → T n satisfyingthe ‘Lipschitz condition’:If y, z ∈ T are adjacent then either ϕ ( y ) = ϕ ( z ) or ϕ ( y ) is adjacent to ϕ ( z ) in T n (in other words, ϕ is a graph homomorphism from T to the graph T (cid:48) n obtained from T n by addinga loop at every vertex). For a domain H ⊂ H , we further set Lip n ( H ) to be the set of ϕ ∈ Lip n satisfying the boundary condition ϕ ( z ) = ρ for all hexagons z which are not in the interior of H (i.e., which are incident to a vertex in V ( H ) \ V ( H )). Define the ‘level lines’ of ϕ ∈ Lip n by ω ϕ := (cid:8) e ∈ E ( H ) : the edge e borders hexagons y, z ∈ T satisfying ϕ ( y ) (cid:54) = ϕ ( z ) (cid:9) . Observe that ω ϕ is a loop configuration and that if ϕ ∈ Lip n ( H ) then ω ϕ ∈ LoopConf ( H, ∅ ). For areal parameter x >
0, define a probability measure ν H,n,x on Lip n ( H ) by ν H,n,x ( ϕ ) := x | ω ϕ | Z Lip
H,n,x , ϕ ∈ Lip n ( H ) , where Z Lip
H,n,x is the unique constant which makes ν H,n,x a probability measure. The definition isextended to x = ∞ by ν H,n, ∞ ( ϕ ) := lim x →∞ ν H,n,x ( ϕ ).The fact that the loop O ( n ) model admits a height function representation is manifested in therelation between the measures ν H,n,x and P ∅ H,n,x . As is straightforward to verify, if ϕ is a randomfunction chosen according to ν H,n,x then ω ϕ is distributed according to P ∅ H,n,x . In particular, theheight function representation of the loop O (1) model is an Ising model (which may be either fer-romagnetic or antiferromagnetic according to whether x < x >
1) and the height functionrepresentation of the loop O (2) model is a restricted Solid-On-Solid model. Our main result, The-orem 1.4, implies that long level lines surrounding a given hexagon are exponentially unlikely inheight functions sampled according to ν H,n,x , when H is a domain of type c ∈ { , , } and n islarge. Our proof does not make use of the height function representation and thus applies to real n . It would be interesting to see whether the height function representation may be used to providefurther information for integer n . Appendix A. Integrals
In this section, we present a detailed derivation of the formulas approximating the partitionfunction and the spin-spin correlations in the spin O ( n ) model on a finite subgraph H of thehexagonal lattice. Let u, v ∈ V ( H ) be distinct vertices and let H + be the (possibly multi-)graphobtained by adding an edge e u,v between u and v to H . In the introductory section, the derivationwas reduced to computing integrals of the form I ( ω ) := (cid:90) Ω (cid:89) { w,w (cid:48) }∈ E ( ω ) (cid:104) σ w , σ w (cid:48) (cid:105) dσ, where Ω = ( √ n · S n − ) V ( H ) , ω is an arbitrary subgraph of H + , and dσ is the product of | V ( H ) | uniform probability measures on √ n · S n − . Note first that, by symmetry, making the substitution σ w ← − σ w for some w ∈ V ( H ) does not change the value of this integral and consequently I ( ω ) = 0unless every vertex has even degree in ω . In other words, if ω ⊂ H then I ( ω ) = 0 unless ω is a loopconfiguration, i.e., ω ∈ LoopConf ( H, ∅ ), and I ( ω + e u,v ) = 0 unless the degrees of u and v in ω areodd and the degrees of all other vertices are even, i.e., ω ∈ LoopConf ( H, ∅ , u, v ).We shall repeatedly make use of the following identity. For every x, y ∈ R n , (cid:90) √ n · S n − (cid:104) x, z (cid:105)(cid:104) z, y (cid:105) dz = (cid:104) x, y (cid:105) , (22)where dz is the uniform probability measure on √ n · S n − . Note that both sides of (22) are bilinearfunctions of x and y and therefore it is enough to verify that (22) holds when x and y are two vectorsfrom the canonical basis { e , . . . , e n } of R n . By symmetry, for each i , (cid:90) √ n · S n − (cid:104) e i , z (cid:105)(cid:104) z, e i (cid:105) dz = 1 n n (cid:88) i =1 (cid:90) √ n · S n − (cid:104) z, e i (cid:105) dz = 1 n (cid:90) √ n · S n − (cid:107) z (cid:107) dz = 1 , If i (cid:54) = j , substituting ( z , . . . , z n ) ← ( z , . . . , z i − , − z i , z i +1 , . . . , z n ) yields (cid:90) √ n · S n − (cid:104) e i , z (cid:105)(cid:104) z, e j (cid:105) dz = − (cid:90) √ n · S n − (cid:104) e i , z (cid:105)(cid:104) z, e j (cid:105) dz = 0 . Suppose first that ω ∈ LoopConf ( H, ∅ ). Since the loops of ω are vertex-disjoint, I ( ω ) = (cid:81) L ⊂ ω I ( L ),where L ranges over all loops of ω . Suppose now that L is a loop through vertices v , . . . , v (cid:96) , where v (cid:96) = v . Invoking (22) repeatedly yields I ( L ) = (cid:90) Ω (cid:104) σ v , σ v (cid:105) · · · (cid:104) σ v (cid:96) − , σ v (cid:96) (cid:105) dσ = (cid:90) Ω (cid:104) σ v , σ v (cid:105) dσ = n, giving I ( ω ) = n L H ( ω ) .Suppose now that ω ∈ LoopConf ( H, ∅ , u, v ), let C be the connected component of u (and v ) in ω ,and note that C must contain a simple path P connecting u and v . Since we have already provedthat I ( L ) = n for every loop L , in order to compute I ( ω + e u,v ), it is enough to compute I ( C + e u,v ).A simple case analysis shows that C is either (i) the path P , (ii) the path P and a loop intersecting P in one of its endpoints, (iii) the path P and two vertex-disjoint loops, each intersecting P in oneof its endpoints, or (iv) the path P and two other simple paths connecting u and v , each pair ofpaths sharing only the vertices u and v . Since the edge e u,v closes P into a loop, invoking (22)repeatedly to ‘contract’ loops yields that I ( C + e u,v ) equals n in case (i), n in case (ii), and n incase (iii). In case (iv), invoking (22) repeatedly only gives I ( C + e u,v ) = (cid:90) (cid:90) √ n · S n − (cid:104) x, y (cid:105) dxdy, XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n which is somewhat more difficult to compute. Using symmetry and the fact that the projection ofthe Lebesgue measure on S n − ⊂ R n onto the first coordinate gives the measure on [ − ,
1] withdensity (1 − t ) n − up to a normalization constant, we obtain I ( C + e u,v ) = (cid:90) √ n · S n − (cid:104) x, √ ne (cid:105) dx = n (cid:90) √ n · S n − (cid:104) x/ √ n, e (cid:105) dx = n · (cid:82) − t (1 − t ) n − dt (cid:82) − (1 − t ) n − dt = 3 n n + 2 , where one may obtain the final identity using integration by parts. Appendix B. Circuits and domains
Here we prove some facts about circuits and domains.
Proof of Fact 2.1.
Let γ be a circuit and denote by H γ the subgraph of H obtained by removingfrom H all edges in γ ∗ . Let Ext( γ ) be the set of vertices that are the endpoint of some infinitesimple path in H γ .First, we claim that Ext( γ ) is a connected component of H γ . To see this, note first that bydefinition, Ext( γ ) is a union of connected components of H γ . Furthermore, since γ ∗ is finite, thereexists an R and a vertex u ∈ V ( H ) such that the complement of the ball of radius R (in the graphdistance determined by H ) centered at u induces the same connected graph H R in both H and H γ . Finally, every infinite simple path in H intersects H R and therefore Ext( γ ) consists of a singleconnected component.Second, we claim that the set of endpoints of the edges in γ ∗ intersects at most two connectedcomponents of H γ , one of which is Ext( γ ). To see this, suppose that γ = ( γ , . . . , γ m ) as in thedefinition in Section 2.1. In order to prove the first part of our claim, it suffices to show that foreach i ∈ { , . . . , m − } , there are two disjoint H γ -connected sets of vertices, each of which intersectsboth { γ i − , γ i } ∗ and { γ i , γ i +1 } ∗ (where we regard an edge as the set of its endpoints). To see this,note that { γ i − , γ i } ∗ and { γ i , γ i +1 } ∗ are the only two out of six edges surrounding the hexagon γ i that belong to γ ∗ . Consequently, the removal of γ ∗ partitions the six vertices surrounding γ i intotwo H γ -connected sets, each of which intersects both { γ i − , γ i } ∗ and { γ i , γ i +1 } ∗ . For the secondpart of the claim, consider an arbitrary infinite simple path in H which uses an edge from γ ∗ . Let { v, w } be the last edge of γ ∗ on this path and observe that either v or w belongs to Ext( γ ). Hence,Ext( γ ) is one of the H γ -connected components that contains an endpoint of an edge of γ ∗ .Third, we claim that Ext( γ ) (cid:54) = V ( H ). If this were not the case, then in particular there would bea { v, w } ∈ γ ∗ such that both v and w belong to the same connected component of H γ . Consequently,there would be a simple path P in H γ that connects v and w . The edge { v, w } and P would thenform a cycle in H that contains exactly one edge of γ ∗ . This is impossible since the basic 6-cyclessurrounding the hexagons of T generate the cycle space of H and each of these basic cycles intersects γ ∗ in either 0 or 2 edges.Fourth, we claim that V ( H ) \ Ext( γ ) is H γ -connected, that is, every two v, w / ∈ Ext( γ ) are in thesame connected component of H γ . To see this, consider two infinite simple paths P v and P w in H that start at v and w , respectively. Since v, w / ∈ Ext( γ ), both P v and P w contain an edge from γ ∗ .Let v (cid:48) , w (cid:48) be the first vertices in P v and P w , respectively, which are incident to edges of γ ∗ . Clearly v, v (cid:48) and w, w (cid:48) lie in the same H γ -connected components, other than Ext( γ ). By our second claim, v (cid:48) and w (cid:48) must belong to the same H γ -connected component. Hence, v and w also belong to thesame H γ -connected component, which we shall from now on denote by Int( γ ).Finally, we show that both Ext( γ ) and Int( γ ), as H γ -connected components, are induced sub-graphs of H and that Int( γ ) is finite. The first assertion follows from the fact that the two endpointsof each edge of γ ∗ belong to different H γ -connected components, which we have already established above. If the second assertion were false, then Int( γ ) would be an infinite H γ -connected graph andhence it would contain an infinite simple path, contradicting the fact that Int( γ ) ∩ Ext( γ ) = ∅ . (cid:3) Proof of Fact 2.2.
Let H be a domain and let E be the set of edges of H with exactly one endpointin V ( H ). Let T be the auxiliary graph with vertex set T whose edges are all pairs { y, z } suchthat { y, z } ∗ ∈ E . We first claim that all vertex degrees in T are even. Indeed, to see this for thedegree of a hexagon z ∈ T , it suffices to traverse the vertices bordering z in order and to considerwhich of them belong to V ( H ). It follows that T contains a circuit γ . By Fact 2.1, γ ∗ splits H intoexactly two connected components. As γ ∗ ⊆ E and Int( γ ) is finite and non-empty, and as V ( H )is finite, connected and with connected complement, it must be that V ( H ) \ V ( H ) ⊆ Ext V ( γ ) and V ( H ) ⊆ Int V ( γ ). Consequently, H = Int( γ ). (cid:3) Proof of Fact 2.3.
Denote A := Int V ( σ ), A (cid:48) := Int V ( σ (cid:48) ) and B := A ∪ A (cid:48) . Let us first show that B is connected. If A ∩ A (cid:48) (cid:54) = ∅ then this is immediate. Otherwise, by assumption, there exists an edge { v, u } ∈ σ ∗ ∩ ( σ (cid:48) ) ∗ . Assume without loss of generality that v ∈ A and u / ∈ A . Then u ∈ A (cid:48) and v / ∈ A (cid:48) , and thus, B is connected.Let C be the unique infinite connected component of V ( H ) \ B and let D := V ( H ) \ C . It isstraightforward to check that D is finite, B ⊂ D and ∂D ⊂ ∂B . Since B is connected, this impliesthat D is connected. Thus, as V ( H ) \ D = C is connected, the subgraph of H induced by D is adomain.By Fact 2.2, there exists a circuit γ such that D = Int V ( γ ). It remains to check that γ ∗ ⊂ σ ∗ ∪ ( σ (cid:48) ) ∗ .Let { v, u } ∈ γ ∗ be such that v ∈ D and u / ∈ D . In particular, v ∈ B and u / ∈ B . Thus, either v ∈ A so that { v, u } ∈ σ ∗ , or v ∈ A (cid:48) so that { v, u } ∈ ( σ (cid:48) ) ∗ . (cid:3) References
1. Michael Aizenman,
Translation invariance and instability of phase coexistence in the two-dimensional Ising system ,Comm. Math. Phys. (1980), no. 1, 83–94. MR 573615 (82c:82003)2. Rodney J. Baxter, Hard hexagons: exact solution , J. Phys. A (1980), no. 3, L61–L70. MR 560533 (80m:82052)3. , Exactly solved models in statistical mechanics , Academic Press Inc. [Harcourt Brace Jovanovich Publish-ers], London, 1989, Reprint of the 1982 original. MR 998375 (90b:82001)4. Vincent Beffara and Hugo Duminil-Copin,
The self-dual point of the two-dimensional random-cluster model iscritical for q ≥
1, Probab. Theory Related Fields (2012), no. 3-4, 511–542. MR 29486855. B´ela Bollob´as,
The art of mathematics , Cambridge University Press, New York, 2006, Coffee time in Memphis.MR 2285090 (2007h:00005)6. Dmitry Chelkak, Hugo Duminil-Copin, Cl´ement Hongler, Antti Kemppainen, and Stanislav Smirnov,
Convergenceof Ising interfaces to Schramm’s SLE curves , C. R. Math. Acad. Sci. Paris (2014), no. 2, 157–161. MR 31518867. Dmitry Chelkak and Stanislav Smirnov,
Universality in the 2D Ising model and conformal invariance of fermionicobservables , Invent. Math. (2012), no. 3, 515–580. MR 29573038. L. Coquille, H. Duminil-Copin, D. Ioffe, and Y. Velenik,
On the Gibbs states of the noncritical Potts model on Z ,Probability Theory and Related Fields (2014), 477–512.9. Eytan Domany, D Mukamel, Bernard Nienhuis, and A Schwimmer, Duality relations and equivalences for modelswith O(n) and cubic symmetry , Nuclear Physics B (1981), no. 2, 279–287.10. C. Domb and M.S. Green,
Phase transitions and critical phenomena , vol. 3, Academic New-York Press, 1976.11. Hugo Duminil-Copin,
Parafermionic observables and their applications to planar statistical physics models , EnsaiosMatematicos, vol. 25, Brazilian Mathematical Society, 2013.12. Hugo Duminil-Copin,
Geometric representations of lattice spin models , Spartacus Graduate, Cours Peccot, Coll`egede France (2015).13. Hugo Duminil-Copin and Stanislav Smirnov,
Conformal invariance of lattice models , Probability and statisticalphysics in two and more dimensions, Clay Math. Proc., vol. 15, Amer. Math. Soc., Providence, RI, 2012, pp. 213–276. MR 302539214. ,
The connective constant of the honeycomb lattice equals (cid:112) √
2, Ann. of Math. (2) (2012), no. 3,1653–1665. MR 291271415. J¨urg Fr¨ohlich and Thomas Spencer,
The Kosterlitz-Thouless transition in two-dimensional Abelian spin systemsand the Coulomb gas , Comm. Math. Phys. (1981), no. 4, 527–602. MR 634447 (83b:82029)16. Geoffrey R Grimmett, The random-cluster model , vol. 333, Springer Science & Business Media, 2006.
XPONENTIAL DECAY OF LOOP LENGTHS IN THE LOOP O ( n ) MODEL WITH LARGE n
17. G. Heller and H.A. Kramers,
Ein Klassisches Modell des Ferromagnetikums und seine nachtr¨agliche Quantisierungim Gebiete tiefer Temperaturen , Ver. K. Ned. Akad. Wetensc.(Amsterdam) (1934), 378–385.18. Y. Higuchi, On the absence of non-translation invariant Gibbs states for the two-dimensional Ising model , Randomfields, Vol. I, II (Esztergom, 1979), Colloq. Math. Soc. J´anos Bolyai, vol. 27, North-Holland, Amsterdam, 1981,pp. 517–534. MR 712693 (84m:82020)19. Richard Kenyon,
An introduction to the dimer model , School and Conference on Probability Theory, ICTP Lect.Notes, XVII, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2004, pp. 267–304 (electronic).20. ,
Conformal invariance of loops in the double-dimer model , Comm. Math. Phys. (2014), no. 2, 477–497.21. D Kim and RI Joseph,
Exact transition temperature of the potts model with q states per site for the triangular andhoneycomb lattices , Journal of Physics C: Solid State Physics (1974), no. 8, L167.22. JM Kosterlitz and DJ Thouless, Ordering, metastability and phase transitions in two-dimensional systems , Journalof Physics C: Solid State Physics (1973), no. 7, 1181–1203.23. R. Koteck´y and S. B. Shlosman, First-order phase transitions in large entropy lattice models , Comm. Math. Phys. (1982), no. 4, 493–515. MR 649814 (83i:82033)24. Antti J. Kupiainen, On the /n expansion , Comm. Math. Phys. (1980), no. 3, 273–294. MR 574175 (81g:82007)25. Lahoussine Laanait, Alain Messager, Salvador Miracle-Sol´e, Jean Ruiz, and Senya Shlosman, Interfaces in thePotts model. I. Pirogov-Sinai theory of the Fortuin-Kasteleyn representation , Comm. Math. Phys. (1991),no. 1, 81–91. MR 1124260 (93d:82021a)26. W. Lenz,
Beitrag zum Verst¨andnis der magnetischen Eigenschaften in festen K¨orpern. , Phys. Zeitschr. (1920),613–615.27. Oliver A. McBryan and Thomas Spencer, On the decay of correlations in
SO( n ) -symmetric ferromagnets , Comm.Math. Phys. (1977), no. 3, 299–302. MR 0441179 (55 Absence of ferromagnetism or antiferromagnetism in one-or two-dimensionalisotropic Heisenberg models , Physical Review Letters (1966), 1133–1136.29. Bernard Nienhuis, Exact Critical Point and Critical Exponents of O( n ) Models in Two Dimensions , PhysicalReview Letters (1982), no. 15, 1062–1065.30. Bernard Nienhuis, Locus of the tricritical transition in a two-dimensional q-state potts model , Physica A: StatisticalMechanics and its Applications (1991), no. 1-3, 109–113.31. J. Palmer,
Planar Ising correlations , Progress in Math. Physics, vol. 49, Birkh¨auser Boston Inc., Boston, MA,2007.32. R. Peierls,
On Ising’s model of ferromagnetism. , Math. Proc. Camb. Phil. Soc. (1936), 477–481.33. A Polyakov, Interaction of goldstone particles in two dimensions. Applications to ferromagnets and massive Yang-Mills fields , Physics Letters B (1975), no. 1, 79–81.34. H.E. Stanley, Dependence of critical properties on dimensionality of spins , Physical Review Letters (1968),no. 12, 589–592.35. DJ Thouless, Long-range order in one-dimensional Ising systems , Physical Review (1969), 732–733.36. V.G. Vaks and A.I. Larkin,
On Phase Transitions of Second Order , Soviet Journal of Experimental and TheoreticalPhysics (1966), 678. Hugo Duminil-CopinUniversit´e de Gen`eve, D´epartement de math´ematiques, 1211 Gen`eve 4, Switzerland.
E-mail address : [email protected] URL : Ron PeledTel Aviv University, School of Mathematical Sciences, Tel Aviv, 69978, Israel.
E-mail address : [email protected] URL : Wojciech SamotijTel Aviv University, School of Mathematical Sciences, Tel Aviv, 69978, Israel.
E-mail address : [email protected] URL : Yinon SpinkaTel Aviv University, School of Mathematical Sciences, Tel Aviv, 69978, Israel.
E-mail address : [email protected] URL ::