Conformally invariant scaling limits in planar critical percolation

This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLE(k)), a one-parameter family of conformally invariant random curves discovered by Schramm (2000). The article is organized around the aim of proving the result, due to Smirnov (2001) and to Camia and Newman (2007), that the percolation exploration path converges in the scaling limit to chordal SLE(6). No prior knowledge is assumed beyond some general complex analysis and probability theory.Comment: 55 pages, 10 figure

Ising model and s-embeddings of planar graphs

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.Comment: 64 pages, 5 figures, minor update of the first version: Eq. (2.13) and Proposition 2.10 added; Lemma 4.2 replaced by Lemma A.