We rigorously prove the existence and the conformal invariance of scaling limits of the magnetization and multi-point spin correlations in the critical Ising model on arbitrary simply connected planar domains. This solves a number of conjectures coming from the physical and the mathematical literature. The proof relies on convergence results for discrete holomorphic spinor observables and probabilistic techniques.

We prove convergence results for variants of Smirnov's fermionic observable in the critical planar Ising model in presence of free boundary conditions. One application of our analysis is a simple proof of a theorem by Hongler and Kytölä on convergence of critical Ising interfaces with plus-minus-free boundary conditions to dipolar SLE(3), and a generalization of this result to an arbitrary number of arcs carrying plus, minus or free boundary conditions. Another application is a computation of scaling limits of crossing probabilities in the critical FK-Ising model with arbitrary number of alternating wired/free boundary arcs. We also deduce a new crossing formula for the spin Ising model.The Stochastic Loewner evolution, introduced by Schramm in [Sch00], is a powerful tool in the study of lattice models in two-dimensional statistical mechanics at criticality. In this approach, one describes random geometric shapes arising in the models by planar growth processes. By means of Loewner's equation for the evolution of conformal maps, such processes can be encoded by continuous, real-valued "driving functions" (see, e. g., [Law05]).Schramm's original idea (often called "Schramm's principle") was that for certain boundary conditions, natural conformal invariance and "domain Markov property" assumptions on a random curve can be restated in terms of its driving function. In the scaling limit, these properties identify the latter as a Brownian motion B κt , where the intensity κ > 0 represents the universality class of the model. This approach, pursued in particular in [Smi01, LSW04, SS05, SS09, CN06, ChSm12, CDHKS13], was extremely fruitful. In a more general setup (e. g. for more complicated boundary conditions), the driving processes are typically described by Brownian motion B κt with time-dependent drifts; these processes do not admit such a simple axiomatic characterization anymore, and a lot of work has been done (see e. g. [LSW03,BBH05,BBK05,LK07,Dub07,Zha08,Dub09,IK13,FK13,KP14]) in order to understand them both in general and in relation to concrete lattice models.One celebrated result in the area is the proof of conformal invariance of fermionic observables in the critical Ising model [Smi06,ChSm12], leading in particular to the proof that interfaces in the model and its random cluster representation converge to SLE 3 and SLE 16 3 respectively [CDHKS13]. This result was extended in [Izy13] to radial and multiple SLE and to multiply-connected domains with suitable analogs of Dobrushin boundary conditions. Another very interesting case, namely that of free boundary conditions, was treated by Hongler and Kytölä [HK13], who proved a conjecture of [BBH05] that interfaces in the critical Ising model on a simply-connected domain with plus-minus-free boundary conditions converge to the dipolar SLE 3 , i. e., the SLE κ (ρ) process [LSW03, SW05] with κ = 3 and ρ = − 3 2 . The beautiful proof of Hongler and Kytölä was quite complicated, the main source of technical difficulties being that they did not use discrete...

We prove a general result on convergence of interfaces in the critical planar Ising model to conformally invariant curves absolutely continuous with respect to SLE(3). Our setup includes multiple interfaces on arbitrary finitely connected domains, and we also treat the radial SLE case. In the case of simply and doubly connected domains, the limiting processes are described explicitly in terms of rational and elliptic functions, respectively. arXiv:1309.5302v2 [math-ph]

Abstract. We introduce a new version of discrete holomorphic observables for the critical planar Ising model. These observables are holomorphic spinors defined on double covers of the original multiply connected domain. We compute their scaling limits, and show their relation to the ratios of spin correlations, thus providing a rigorous proof to a number of formulae for those ratios predicted by CFT arguments.