Ising model and s-embeddings of planar graphs

Chelkak, Dmitry

doi:10.48550/arxiv.2006.14559

Cited by 5 publications

(27 citation statements)

References 36 publications

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“…Moreover, this Cauchy formula can be written in a purely abstract form without any assumption on the embedding or weights of the model under consideration (see Lemma 3.14), which paves the way for further generalizations of the convergence results for spin correlations beyond the Z-invariant setup; cf. recent results on the convergence of fermionic observables on the so-called s-embeddings of planar weighted graphs [12].…”

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confidence: 99%

“…Another -though not strictly necessary for our analysis -new idea implemented in this paper is a re-embedding of the massive Z-invariant Ising model on Λ δ into the complex plane using the aforementioned s-embeddings S δ ; see Section 3.3 for more details. This allows us to benefit from a general regularity theory developed for s-holomorphic functions on s-embeddings in [12,Section 2] and [18,Section 6]. Under this procedure, the original massive s-holomorphic observables on Λ δ and new s-holomorphic observables on S δ are linked by a simple explicit formula given in Proposition 3.21, which immediately allows us to deduce the a priori regularity of massive s-holomorphic functions on Λ δ from the results of [12,18].…”

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confidence: 99%

“…This allows us to benefit from a general regularity theory developed for s-holomorphic functions on s-embeddings in [12,Section 2] and [18,Section 6]. Under this procedure, the original massive s-holomorphic observables on Λ δ and new s-holomorphic observables on S δ are linked by a simple explicit formula given in Proposition 3.21, which immediately allows us to deduce the a priori regularity of massive s-holomorphic functions on Λ δ from the results of [12,18]. Also, this provides a concrete illustration of a general phenomenology, which says that the mass in a planar Ising model manifests itself as the mean curvature of an s-embedding of the model into the Minkowski space R 2+1 ; see [12,Section 2.7] for a discussion.…”

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Universality of spin correlations in the Ising model on isoradial graphs

Chelkak¹,

Izyurov²,

Mahfouf³

2021

Preprint

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We prove universality of spin correlations in the scaling limit of the planar Ising model on isoradial graphs with uniformly bounded angles and Z-invariant weights. Specifically, we show that in the massive scaling limit, i. e., as the mesh size δ tends to zero at the same rate as the inverse temperature goes to the critical one, the two-point spin correlations in the full plane behave aswhere the universal constant Cσ and the function Ξ(|u 1 − u 2 |, m) are independent of the lattice. The mass m is defined by the relation k − 1 ∼ 4mδ, where k is the Baxter elliptic parameter. This includes m of both signs as well as the critical case when Ξ(r, 0) = r −1/4 .These results, together with techniques developed to obtain them, are sufficient to extend to isoradial graphs the convergence of multi-point spin correlations in finite planar domains on the square grid, which was established in a joint work of the first two authors and C. Hongler at criticality, and by S. C. Park in the sub-critical massive regime. We also give a simple proof of the fact that the infinite-volume magnetization in the Z-invariant model is independent of the site and of the lattice.As compared to techniques already existing in the literature, we streamline the analysis of discrete (massive) holomorphic spinors near their ramification points, relying only upon discrete analogues of the kernel z −1/2 for m = 0 and of z −1/2 e ±2m|z| for m = 0. Enabling the generalization to isoradial graphs and providing a solid ground for further generalizations, our approach also considerably simplifies the proofs in the square lattice setup.

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Universality of spin correlations in the Ising model on isoradial graphs

Chelkak¹,

Izyurov²,

Mahfouf³

2021

Preprint

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“…When the context is clear, we will drop the denomination corner or diamond s-holomorphic functions and just call them s-holomorphic functions, passing from one to the other using Proposition 4.12. As a consequence of the projections equality above mentioned, one can see as in [Che20,(2.20) and Remark 2.9] that F satisfies the maximum principle, i.e. for any connected set S of ♦(G) (where neighbours are either at a distance 2δ from each other or belong to same vertical axis), the maximum of |F | is attained at the boundary of S.…”

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confidence: 90%

“…[CIM21]). It could be even possible in principle to bypass the use of all sub/super harmonicity properties using ideas of [Che20].…”

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confidence: 99%

Conformal invariance in the quantum Ising model

Li¹,

Mahfouf²

2021

Preprint

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We introduce Kadanoff-Ceva order-disorder operators in the quantum Ising model. This approach was first used for the classical planar Ising model [KC71] and recently put back to the stage [CCK17,Che18]. This representation turns out to be equivalent to the loop expansion of Sminorv's fermionic observables [Smi10] and is particularly interesting due to its simple and compact formulation. Using this approach, we are able to extend different results known in the classical planar Ising model, such as the conformal invariance / covariance of correlations and the energy-density, to the spin-representation of the (1+1)-dimensional quantum Ising model.

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Dimer model and holomorphic functions on t‐embeddings of planar graphs

Chelkak¹,

Laslier

Russkikh

2023

Proceedings of London Math Soc

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We introduce the framework of discrete holomorphic functions on t-embeddings of weighted bipartite planar graphs; t-embeddings also appeared under the name Coulomb gauges in a recent paper (Kenyon, Lam, Ramassamy, and Russkikh, Dimers and circle patterns, 2018). We argue that this framework is particularly relevant for the analysis of scaling limits of the height fluctuations in the corresponding dimer models. In particular, it unifies both Kenyon's interpretation of dimer observables as derivatives of harmonic functions on T-graphs and the notion of s-holomorphic functions originated in Smirnov's work on the critical Ising model. We develop an a priori regularity theory for such functions and provide a meta-theorem on convergence of the height fluctuations to the Gaussian Free Field. We also discuss how several more standard discretizations of complex analysis fit this general framework.M S C 2 0 2 0 82B20, 30G25 (primary) Dmitry Chelkak, on leave from St.

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Ising model and s-embeddings of planar graphs

Cited by 5 publications

References 36 publications

Universality of spin correlations in the Ising model on isoradial graphs

Universality of spin correlations in the Ising model on isoradial graphs

Conformal invariance in the quantum Ising model

Dimer model and holomorphic functions on t‐embeddings of planar graphs

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