Grounded Lipschitz functions on trees are typically flat

Peled, Ron; Samotij, Wojciech; Yehudayoff, Amir

doi:10.1214/ecp.v18-2796

Cited by 10 publications

(10 citation statements)

References 8 publications

(15 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Galvin and Engbers [5] established the analog of the conjecture, and more general rigidity results for graph homomorphisms, in the limit when n is fixed and d tends to infinity. Similar rigidity results on expander and tree graphs are established in [21,22,23].…”

Section: Background and Related Worksupporting

confidence: 72%

See 1 more Smart Citation

Rigidity of 3-colorings of the discrete torus

Feldheim¹,

Peled²

2018

Ann. Inst. H. Poincaré Probab. Statist.

Self Cite

View full text Add to dashboard Cite

We prove that a uniformly chosen proper 3-coloring of the d-dimensional discrete torus has a very rigid structure when the dimension d is sufficiently high. We show that with high probability the coloring takes just one color on almost all of either the even or the odd sub-torus. In particular, one color appears on nearly half of the torus sites. This model is the zero temperature case of the 3-state anti-ferromagnetic Potts model from statistical physics.Our work extends previously obtained results for the discrete torus with specific boundary conditions. The main challenge in this extension is to overcome certain topological obstructions which appear when no boundary conditions are imposed on the model. Locally, a proper 3-coloring defines the discrete gradient of an integer-valued height function which changes by exactly one between adjacent sites. However, these locally-defined functions do not always yield a height function on the entire torus, as the gradients may accumulate to a non-zero quantity when winding around the torus. Our main result is that in high dimensions, a global height function is well defined with high probability, allowing to deduce the rigid structure of the coloring from previously known results. Moreover, the probability that the gradients accumulate to a vector m, corresponding to the winding in each of the d directions, is at most exponentially small in the product of m ∞ and the area of a cross-section of the torus.In the course of the proof we develop discrete analogues of notions from algebraic topology. This theory is developed in some generality and may be of use in the study of other models.

show abstract

Section: Background and Related Worksupporting

confidence: 72%

“…The translation trichotomy, Theorem 3.4, and the relations ( 21) and (22) imply that Type(U ) = 1, establishing (19). Now observe that U c , V c are also translation respecting and satisfy V c ⊆ U c .…”

Section: Corollaries Of the Trichotomymentioning

confidence: 82%

Rigidity of 3-colorings of the discrete torus

Feldheim¹,

Peled²

2018

Ann. Inst. H. Poincaré Probab. Statist.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Observe that any large component B of X satisfies |∂B| ≥ d 2 (by Corollary 2.3b) and ∂B ⊂ ∂Y . Therefore, by (36) and (37),…”

Section: 2mentioning

confidence: 88%

Long-range order in the 3-state antiferromagnetic Potts model in high dimensions

Feldheim

Spinka

2019

J. Eur. Math. Soc.

View full text Add to dashboard Cite

We prove the existence of long-range order for the 3-state Potts antiferromagnet at low temperature on Z d for sufficiently large d. In particular, we show the existence of six extremal and ergodic infinite-volume Gibbs measures, which exhibit spontaneous magnetization in the sense that vertices in one bipartition class have a much higher probability to be in one state than in either of the other two states. This settles the high-dimensional case of the Kotecký conjecture. IntroductionThe q-state Potts model is a classical model in statistical mechanics, which generalizes the Ising model by allowing more than two states. A special case was first considered by Ashkin and Teller in 1943 [1], while the general model was proposed by Domb and published by Potts in 1951 [38]. Since the late 1970's, the model has drawn substantial attention from mathematicians and physicists alike, largely because it proved to be very rich, displaying a much wider spectrum of phenomena than the simpler Ising model. For an extensive survey of classical results on the Potts model, see Wu [46].While the ferromagnetic regime of the model is by now relatively well understood, the picture for the antiferromagnetic regime is far from complete. Here we consider the 3-state antiferromagnetic (AF) Potts model on the integer lattice Z d . The model assigns a random value f (v) ∈ {0, 1, 2} to each vertex v in some domain Λ ⊂ Z d , favoring different states on adjacent vertices. The probability of any given configuration f is proportional to exp(−βH f ), where β ≥ 0 is a real parameter andwhere the sum is taken over all pairs of nearest neighbors. In other words, f is distributed according to the Boltzmann distribution with the Hamiltonian H f at inverse temperature β.At infinite temperature (β = 0), the values assigned to different vertices are independent, and the model is completely disordered. The Dobrushin uniqueness condition [8] guarantees that, in any dimension, disorder persists at sufficiently high temperature (small β). A fundamental question is whether at low temperature (large β) the model remains disordered or, instead, undergoes a phase transition into an ordered phase. In the latter case, it is also desirable to understand the structure of a typical ordered configuration. Such a phase transition does not occur in every dimension (e.g., for d = 1), and it is interesting to determine in which dimensions (if any) it does.Following an earlier debate in the physical community (see e.g.[2] where a continuous transition was conjectured), Kotecký conjectured circa 1985 (implied in [29], also mentioned in [18]) that in high dimensions, possibly already in three dimensions, the 3-state AF Potts model indeed undergoes a phase transition, and that at sufficiently low temperature, a configuration typically follows one of six patterns. To understand the nature of these patterns, note first that the graph Z d is bipartite, and that each bipartition class forms a sublattice. In a typical large-volume disordered configuration, Thus, µ τ Λ,∞ is the u...

show abstract

“…These may be used to prove bounds on the tail of 1 To the best of our knowledge this is the first instance when a uniformly distributed Lipschitz function is proven to have logarithmically diverging variance. Previously known results establish that the variance is bounded (referred to as localisation) in high dimensions [32], or when the underlying graph is a tree [33] or an expander [34]. The conjectured convergence of the height function to the GFF indicates that localisation should also hold on lattices in dimensions three and above.…”

Section: Uniform Lipschitz Functionsmentioning

confidence: 93%

Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations

Glazman

Manolescu

2021

Commun. Math. Phys.

View full text Add to dashboard Cite

Uniform integer-valued Lipschitz functions on a domain of size N of the triangular lattice are shown to have variations of order $$\sqrt{\log N}$$ log N . The level lines of such functions form a loop O(2) model on the edges of the hexagonal lattice with edge-weight one. An infinite-volume Gibbs measure for the loop O(2) model is constructed as a thermodynamic limit and is shown to be unique. It contains only finite loops and has properties indicative of scale-invariance: macroscopic loops appearing at every scale. The existence of the infinite-volume measure carries over to height functions pinned at the origin; the uniqueness of the Gibbs measure does not. The proof is based on a representation of the loop O(2) model via a pair of spin configurations that are shown to satisfy the FKG inequality. We prove RSW-type estimates for a certain connectivity notion in the aforementioned spin model.

show abstract

Grounded Lipschitz functions on trees are typically flat

Cited by 10 publications

References 8 publications

Rigidity of 3-colorings of the discrete torus

Rigidity of 3-colorings of the discrete torus

Long-range order in the 3-state antiferromagnetic Potts model in high dimensions

Uniform Lipschitz Functions on the Triangular Lattice Have Logarithmic Variations

Contact Info

Product

Resources

About