Circle Patterns and Critical Ising Models

Lis, Marcin

doi:10.1007/s00220-019-03541-1

Cited by 13 publications

(11 citation statements)

References 39 publications

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“…Remark 6.4. Though this is not fully clear at the moment, we hope that, at least in some situations of interest (e.g., see Section 6.4 or[38]), s-subharmonic functions H F obtained via (2.14) are a priori close to s-harmonic ones; recall that this is exactly the viewpoint developed in[18, Section 3] for the critical Z-invariant model. Also, note that extending the domain of definition of H F to ♦(G) similarly to (6.1), one can easily see that thus obtained functions satisfy the maximum principle.…”

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confidence: 84%

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Planar Ising Model at Criticality: State-of-the-Art and Perspectives

Chelkak

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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In this essay, we briefly discuss recent developments, started a decade ago in the seminal work of Smirnov and continued by a number of authors, centered around the conformal invariance of the critical planar Ising model on Z 2 and, more generally, of the critical Z-invariant Ising model on isoradial graphs (rhombic lattices). We also introduce a new class of embeddings of general weighted planar graphs (s-embeddings), which might, in particular, pave the way to true universality results for the planar Ising model.2010 Mathematics Subject Classification. Primary 82B20; Secondary 30G25, 60J67, 81T40.

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confidence: 84%

“…Below we only sketch some important features of the construction, details will appear elsewhere. Independently of our paper, a special class of s-embeddings -circle patterns -was studied by Lis in [38] and the criticality was proven in the case of uniformly bounded faces.…”

Section: Towards Universality Beyond Isoradial Graphsmentioning

confidence: 99%

Planar Ising Model at Criticality: State-of-the-Art and Perspectives

Chelkak

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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“…We refer the interested reader to [43,Section 4] for more references on 'the relationship between the Ising model and discrete holomorphy' and for a discussion of special solutions (Dirac spinors) of the propagation equation (1.5) on discrete Riemann surfaces, and to [6] for more information on the non-critical model. • Critical Ising model on circle patterns introduced by Lis in [41] as a generalization of the isoradial context. In this setup the quads S (z) are kites, which allow to view S as a circle pattern.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

Ising model and s-embeddings of planar graphs

Chelkak¹

2020

Preprint

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We introduce the notion of s-embeddings S = S F of planar graphs carrying a (critical) nearest-neighbor Ising model; the construction is based upon a choice of a special solution F of the three-terms propagation equation for Kadanoff-Ceva fermions, a so-called Dirac spinor. Each Dirac spinor F provides an interpretation of all other solutions of the propagation equations as sholomorphic functions on the s-embedding S F , the notion of s-holomorphicity generalizes Smirnov's definition [50] on the square grid (as well as on isoradial graphs) and is a special case of t-holomorphic functions on t-embeddings appearing in the bipartite dimer model context.We set up a general framework for the analysis of s-holomorphic functions on s-embeddings S δ with δ → 0 (algebraic identities, a priori regularity theory etc) and then focus on the simplest situation when S δ satisfy the uniformly bounded lengths/angles assumptions and also lead to the horizontal (more precisely, O(δ)) profiles of the associated functions Q δ ; the latter can be viewed as the origami maps associated to S δ in the dimer model terminology. A very particular case when all these assumptions hold is provided by the critical Ising model on a doubly-periodic graph under its canonical s-embedding, another example is the critical Ising model on circle patterns.Under these assumptions on S δ and Q δ we prove the convergence of basic fermionic observables, similar to those from [50], to a conformally covariant limit; it is worth noting that the strategy of the proof is different from that used in [50] due to the lack of tools specific for the isoradial setup. Together with the RSW-type crossing estimates, which we prove under the same assumptions, this also implies the convergence of interfaces in the random cluster representation of the Ising model to Schramm's SLE(16/3) curves.

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“…As mentioned in the introduction, there are two notable classes of examples of α-embeddings. The class of 1-embeddings of a planar graph correspond to s-embeddings defined by Chelkak in [5] (see also [19]), while the class of 2embeddings correspond to harmonic embeddings [22,13].…”

Section: Embeddings and Realizationsmentioning

confidence: 99%

Cube moves for $s$-embeddings and $α$-realizations

Melotti¹,

Ramassamy²,

Thévenin³

2020

Preprint

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For every α ∈ R * , we introduce the class of α-embeddings as tilings of a portion of the plane by quadrilaterals such that the side-lengths of each quadrilateral ABCD satisfy AB α + CD α = AD α + BC α . When α is 1 (resp. 2) we recover the so-called s-embeddings (resp. harmonic embeddings). We study existence and uniqueness properties of a local transformation of α-embeddings (and of more general α-realizations, where the quadrilaterals may overlap) called the cube move, which consists in flipping three quadrilaterals that meet at a vertex, while staying within the class of α-embeddings. The special case α = 1 (resp. α = 2) is related to the star-triangle transformation for the Ising model (resp. for resistor networks). In passing, we give a new and simpler formula for the change in coupling constants for the Ising star-triangle transformation.

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Circle Patterns and Critical Ising Models

Cited by 13 publications

References 39 publications

Planar Ising Model at Criticality: State-of-the-Art and Perspectives

Planar Ising Model at Criticality: State-of-the-Art and Perspectives

Ising model and s-embeddings of planar graphs

Cube moves for $s$-embeddings and $α$-realizations

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