On the transition between the disordered and antiferroelectric phases of the 6-vertex model

Glazman, Alexander; Peled, Ron

doi:10.48550/arxiv.1909.03436

Cited by 9 publications

(23 citation statements)

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“…The staggered model and the standard one actually agree when a = b. It has been proved that in this case the height function is localized for a < 1/2 [19], and it is expected that it delocalizes for all a ≥ 1/2. The only rigorously solved cases in this regime are so far a = 1/2 [19], a = √ 2/2 [9,25] and its small neighbourhood [18], and a = 1 [7].…”

Section: Introductionmentioning

confidence: 63%

“…In the latter case it is shown that the height function delocalizes, i.e., has unbounded variance as the system grows. This was first proved (for different boundary conditions) in [19],…”

Section: Introductionmentioning

confidence: 81%

“…As a result, we give a unified framework for some of the known relations between these models [1,6,9,15,19,23,30,31,35,37,39,40].…”

Section: Introductionmentioning

confidence: 99%

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Spins, percolation and height functions

Lis¹

2019

Preprint

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To highlight certain similarities in combinatorial representations of several well known two-dimensional models of statistical mechanics, we introduce and study a new family of models which specializes to these cases after a proper tuning of the parameters.To be precise, our model consists of two independent standard Potts models, with possibly different numbers of spins and different coupling constants (the four parameters of the model), defined jointly on a graph embedded in a surface and its dual graph, and conditioned on the event that the primal and dual interfaces between spins of different value do not intersect. We also introduce naturally related height function and bond percolation models, and we discuss their basic properties and mutual relationship.As special cases we recover the standard Potts and random cluster model, the six-vertex model and loop O(n) model, the random current, double random current and XOR-Ising model.

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Section: Introductionmentioning

confidence: 63%

“…In the latter case it is shown that the height function delocalizes, i.e., has unbounded variance as the system grows. This was first proved (for different boundary conditions) in [19],…”

Section: Introductionmentioning

confidence: 81%

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Spins, percolation and height functions

Lis¹

2019

Preprint

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“…Contrary to the local quartic part, the two terms in the right side of (29) have no reason to cancel. In order to control their behavior under iterations of the RG, we need to exhibit cancellations, as discussed in the next item.…”

Section: Sketch Of the Proofmentioning

confidence: 99%

“…There are a few notable exceptions to this picture: I refer to recent results obtained by this approach on non-exactly solved models and/or models away from the free Fermi point and, more specifically on: crossing probabilities for critical percolation on triangular lattice[37]; Pfaffian nature of boundary spin correlations in interacting Ising models[1]; limit shapes and surface tension for the 5V model[17]; height (de)localization transition in the 6V model[29].…”

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confidence: 99%

Conformal invariance and Renormalization Group

Giuliani¹

2020

Preprint

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Conformal field theory (CFT) is an extremely powerful tool for explicitly computing critical exponents and correlation functions of statistical mechanics systems at a second order phase transition, or of condensed matter systems at a quantum critical point. Conformal invariance is expected to be a feature of the fixed point theory obtained from a microscopic model at criticality, under appropriate averaging and rescaling operations: the action of the Wilsonian Renormalization Group (RG). Unfortunately, an explicit connection between critical microscopic models and their conformally invariant scaling limit is still lacking in general. Nevertheless, the last decades witnessed significant progress on this topic, both from the mathematical and physics sides, where several new tools have been introduced and their ranges of applications have constantly and significantly increased: I refer here, e.g., to discrete holomorphicity, SLE, the use of lattice Ward Identities in constructive RG, the conformal bootstrap program and its recent applications to 3D CFT. In an effort to make further progress on these problems, the one-day workshop Emergent CFTs in statistical mechanics was organized and held at Institut Curie in Paris on January 29, 2020: the goal was to bring together probabilists, mathematical physicists and theoretical physicists, working on various aspects of critical statistical mechanics systems with complementary tools, both at the discrete and the continuum level, in the hope of creating new connections between the different approaches. This paper is based on an introductory talk given at the workshop: after a summary of the main topics discussed in the meeting, I illustrate the approach to the problem based on constructive RG methods, by reviewing recent results on the existence and the explicit characterization of the scaling limit of critical 2D Ising models with finite range interactions in cylindrical geometry.

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Finitary codings for gradient models and a new graphical representation for the six‐vertex model

Ray

Spinka

2021

Random Struct Algorithms

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It is known that the Ising model on Z 𝑑 at a given temperature is a finitary factor of an i.i.d. process if and only if the temperature is at least the critical temperature. Below the critical temperature, the plus and minus states of the Ising model are distinct and differ from one another by a global flip of the spins. We show that it is only this global information which poses an obstruction to being finitary by showing that the gradient of the Ising model is a finitary factor of i.i.d. at all temperatures. As a consequence, we deduce a volume-order large deviation estimate for the energy. Results in the same spirit are shown for the Potts model, the so-called beach model, and the six-vertex model. We also introduce a coupling between the six-vertex model with c ≥ 2 and a new Edwards-Sokal type graphical representation of it, which we believe is of independent interest.

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On the transition between the disordered and antiferroelectric phases of the 6-vertex model

Cited by 9 publications

References 0 publications

Spins, percolation and height functions

Spins, percolation and height functions

Conformal invariance and Renormalization Group

Finitary codings for gradient models and a new graphical representation for the six‐vertex model

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