Ising model and s-embeddings of planar graphs

We discuss the notion of s-embeddings $\mathcal{S}=\mathcal{S}_\mathcal{X}$ of planar graphs carrying a nearest-neighbor Ising model. The construction of $\mathcal{S}_\mathcal{X}$ is based upon a choice of a global complex-valued solution $\mathcal{X}$ of the propagation equation for Kadanoff-Ceva fermions. Each choice of $\mathcal{X}$ provides an interpretation of all other fermionic observables as s-holomorphic functions on $\mathcal{S}_\mathcal{X}$. We set up a general framework for the analysis of such functions on s-embeddings $\mathcal{S}^\delta$ with $\delta\to 0$. Throughout this analysis, a key role is played by the functions $\mathcal{Q}^\delta$ associated with $\mathcal{S}^\delta$, the so-called origami maps in the bipartite dimer model terminology. In particular, we give an interpretation of the mean curvature of the limit of discrete surfaces $(\mathcal{S}^\delta;\mathcal{Q}^\delta)$ viewed in the Minkowski space $\mathbb R^{2,1}$ as the mass in the Dirac equation describing the continuous limit of the model. We then focus on the simplest situation when $\mathcal{S}^\delta$ have uniformly bounded lengths/angles and $\mathcal{Q}^\delta=O(\delta)$; as a particular case this includes all critical Ising models on doubly periodic graphs via their canonical s-embeddings. In this setup we prove RSW-type crossing estimates for the random cluster representation of the model and the convergence of basic fermionic observables. The proof relies upon a new strategy as compared to the already existing literature, it also provides a quantitative estimate on the speed of convergence.