Spins, percolation and height functions

Lis, Marcin

doi:10.48550/arxiv.1909.07351

Cited by 3 publications

(4 citation statements)

References 33 publications

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“…Moreover, by Proposition 7 and Lemma 8, the random loops under P G,β carry probabilistic information about both the spin XY model (in terms of correlation functions) and its dual height function (as an exact coupling). An analogous role for the Ising and Ashkin-Teller model is played by the (double) random current measure that encodes both an integer-valued height function and the spin correlations [10,25,26]. The difference is that for the XY model, the correlations are determined by loop connectivities instead of percolation connectivities.…”

Section: Remarkmentioning

confidence: 99%

An Elementary Proof of Phase Transition in the Planar XY Model

Engelenburg

Lis

2022

Commun. Math. Phys.

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Using elementary methods we obtain a power-law lower bound on the two-point function of the planar XY spin model at low temperatures. This was famously first rigorously obtained by Fröhlich and Spencer (Commun Math Phys 81(4):527–602, 1981) and establishes a Berezinskii–Kosterlitz–Thouless phase transition in the model. Our argument relies on a new loop representation of spin correlations, a recent result of Lammers (Probab Relat Fields, 2021) on delocalisation of general integer-valued height functions, and classical correlation inequalities.

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Section: Remarkmentioning

confidence: 99%

An Elementary Proof of Phase Transition in the Planar XY Model

Engelenburg

Lis

2022

Commun. Math. Phys.

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“…Finally, an extension of this coupling to the Ashkin-Teller model can be found in the works [51,52] that appeared before but were based on the current article. Here we will provide a different proof that uses the associated dimer model representation.…”

Section: A Coupling Between the Double Random Current And The Xor-isi...mentioning

confidence: 90%

“…We leave it to the interested reader to check that the resulting coupling of the primal and dual double random current model is the same as the one described in [52] (where no connection with the dimer model is used).…”

Section: Alternating Flows On G and Dimers On G Dmentioning

confidence: 99%

Conformal invariance of double random currents I: identification of the limit

Duminil-Copin¹,

Lis²,

Qian³

2021

Preprint

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This is the first of two papers devoted to the proof of conformal invariance of the critical double random current and the XOR-Ising models on the square lattice. More precisely, we show the convergence of loop ensembles obtained by taking the cluster boundaries in the sum of two independent currents with free and wired boundary conditions, and in the XOR-Ising models with free and plus/plus boundary conditions. Therefore we establish Wilson's conjecture on the XOR-Ising model [81]. The strategy, which to the best of our knowledge is different from previous proofs of conformal invariance, is based on the characterization of the scaling limit of these loop ensembles as certain local sets of the Gaussian Free Field. In this paper, we identify uniquely the possible subsequential limits of the loop ensembles. Combined with [28], this completes the proof of conformal invariance.

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“…For the six-vertex model (Theorem 3), we construct a similar representation and prove the necessary monotonicity properties and uniqueness of Gibbs measure for large c. This allows us to apply our general result. A short time after the first draft of this paper appeared online, it came to our attention that Marcin Lis [36] had independently obtained a similar representation for a wider class of models.…”

Section: Outline Of Proofmentioning

confidence: 99%

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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Spins, percolation and height functions

Cited by 3 publications

References 33 publications

An Elementary Proof of Phase Transition in the Planar XY Model

An Elementary Proof of Phase Transition in the Planar XY Model

Conformal invariance of double random currents I: identification of the limit

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

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