Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations

Glazman, Alexander; Manolescu, Ioan

doi:10.1007/s00220-020-03920-z

Cited by 10 publications

(8 citation statements)

References 34 publications

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“…The authors of the present paper proved [28] the uniqueness of the Gibbs measure at n = 2 and x = 1, a result which is not covered by Theorem 1.9. However, it is still unproved in this case that the weak limit of any converging sequence of finite volume measures is Gibbs.…”

Section: Loop O(n) Modelmentioning

confidence: 73%

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Structure of Gibbs measures for planar FK-percolation and Potts models

Glazman,

Manolescu

2021

Preprint

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We prove that all Gibbs measures of the q-state Potts model on Z 2 are linear combinations of the extremal measures obtained as thermodynamic limits under free or monochromatic boundary conditions. In particular all Gibbs measures are invariant under translations. This statement is new at points of first-order phase transition, that is at T = T c (q) when q > 4. In this case the structure of Gibbs measures is the most complex in the sense that there exist q + 1 distinct extremal measures.Most of the work is devoted to the FK-percolation model on Z 2 with q ≥ 1, where we prove that every Gibbs measure is a linear combination of the free and wired ones. The arguments are non-quantitative and follow the spirit of the seminal works of Aizenman [1] and Higuchi [32], which established the Gibbs structure for the two-dimensional Ising model. Infinite-range dependencies in FK-percolation (i.e., a weaker spatial Markov property) pose serious additional difficulties compared to the case of the Ising model. For example, it is not automatic, albeit true, that thermodynamic limits are Gibbs. The result for the Potts model is then derived using the Edwards-Sokal coupling and auto-duality. The latter ingredient is necessary since applying the Edwards-Sokal procedure to a Gibbs measure for the Potts model does not automatically produce a Gibbs measure for FK-percolation.Finally, the proof is generic enough to adapt to the FK-percolation and Potts models on the triangular and hexagonal lattices and to the loop O(n) model in the range of parameters for which its spin representation is positively associated.

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Section: Loop O(n) Modelmentioning

confidence: 73%

“…At the moment, the phase diagram is understood only in several regions of parameters. For recent results on the two types of behaviour, we direct the reader to the recent works [10,16,28,27].…”

Section: Monotonic Propertiesmentioning

confidence: 99%

Structure of Gibbs measures for planar FK-percolation and Potts models

Glazman,

Manolescu

2021

Preprint

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“…In addition, for sufficiently large n, exponential decay was proven for all x > 0 (and an ordering transition was further established) [14]. Lastly, existence of macroscopic loops (as well as Russo-Seymour-Welsh type estimates) was recently shown to occur on the line x = x c (n) for 1 ≤ n ≤ 2 [12] and also at n = 2, x = 1 [22] (uniform Lipschitz functions).…”

Section: Exponential Decaymentioning

confidence: 87%

“…Previous techniques for showing the existence of long loops relied on positive association (FKG) properties for a suitable spin representation; such representations are only known to exist in the regimes n ≥ 1, x ≤ 1 √ n [12] and n ≥ 2, x ≤ 1 √ n−1 [22]. Among the merits of the new technique is that it applies in the absence of such FKG properties.…”

Section: Exponential Decaymentioning

confidence: 99%

“…In this section, we will prove Theorem 7 through an expansion of the loop O(n) model in n, which was first used in [8]. A similar expansion was used in [22] and [21]. Once this expansion is established, a straightforward monotonicity argument, as well as simple geometric considerations on the torus, will yield the desired lower bound on the probability of finding a non-contractible loop.…”

Section: Proof Of Theorem 4 Corollary 5 and Corollarymentioning

confidence: 99%

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Macroscopic loops in the loop~$O(n)$ model via the XOR trick

Crawford¹,

Glazman²,

Harel³

et al. 2020

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The loop O(n) model is a family of probability measures on collections of nonintersecting loops on the hexagonal lattice, parameterized by (n, x), where n is a loop weight and x is an edge weight. Nienhuis predicts that, for 0 ≤ n ≤ 2, the model exhibits two regimes: one with short loops when x < x c (n), and another with macroscopic loops when x ≥ x c (n), where x c (n) = 1 2 + √ 2 − n. In this paper, we prove three results regarding the existence of long loops in the loop O(n) model. Specifically, we show that, for some δ > 0 and anywe can conclude the loops are macroscopic. Next, we prove the existence of loops whose diameter is comparable to that of a finite domain whenever n = 1, x ∈ (1, √ 3]; this regime is equivalent to part of the antiferromagnetic regime of the Ising model on the triangular lattice. Finally, we show the existence of non-contractible loops on a torus when n ∈ [1, 2], x = 1.The main ingredients of the proof are: (i) the 'XOR trick': if ω is a collection of short loops and Γ is a long loop, then the symmetric difference of ω and Γ necessarily includes a long loop as well; (ii) a reduction of the problem of finding long loops to proving that a percolation process on an auxiliary graph, built using the Chayes-Machta and Edwards-Sokal geometric expansions, has no infinite connected components; and (iii) a recent result on the percolation threshold of Benjamini-Schramm limits of planar graphs.

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Height function delocalisation on cubic planar graphs

Lammers

2021

Probab. Theory Relat. Fields

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Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations

Cited by 10 publications

References 34 publications

Structure of Gibbs measures for planar FK-percolation and Potts models

Structure of Gibbs measures for planar FK-percolation and Potts models

Macroscopic loops in the loop~$O(n)$ model via the XOR trick

Height function delocalisation on cubic planar graphs

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