Ising model and s-embeddings of planar graphs

Dmitry Chelkak

Published 2020-06-25Version 1

We introduce the notion of s-embeddings $\mathcal{S}=\mathcal{S}_\mathcal{F}$ of planar graphs carrying a (critical) nearest-neighbor Ising model; the construction is based upon a choice of a special solution $\mathcal{F}$ of the three-terms propagation equation for Kadanoff-Ceva fermions, a so-called Dirac spinor. Each Dirac spinor $\mathcal{F}$ provides an interpretation of all other solutions of the propagation equations as s-holomorphic functions on the s-embedding $\mathcal{S}_\mathcal{F}$, the notion of s-holomorphicity generalizes Smirnov's definition on the square grid/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 $\mathcal{S}^\delta$ with $\delta\to 0$ (algebraic identities, a priori regularity theory etc) and then focus on the simplest situation when $\mathcal{S}^\delta$ have uniformly bounded lengths/angles and also lead to the horizontal (more precisely, $O(\delta)$) profiles of the associated functions $\mathcal{Q}^\delta$; the latter can be viewed as the origami maps associated to $\mathcal{S}^\delta$ 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 we prove the convergence of basic fermionic observables to a conformally covariant limit; note that we develop a new strategy of the proof because of 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.