Smirnov’s Observable for Free Boundary Conditions, Interfaces and Crossing Probabilities

Abstract

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...

Planar Ising Model at Criticality: State-of-the-Art and Perspectives

“…Since the functions F δ ̟ are discrete holomorphic, their limit f ̟ is a holomorphic function and one also has h ̟ = Im[(f ̟ (z)) 2 dz]. Moreover, one can show that the boundary conditions (3.7) survive as δ → 0, see [18,Remark 6.3] and [31]. Clearly, h ̟ also inherits from H δ ̟ the semi-boundedness from below near v p and from above near u q .…”

“…Even more importantly, H F encodes the boundary conditions of the Ising model in the form of Dirichlet boundary conditions. Namely, one can choose an additive constant in its definition so that where ∂ νin stands for a discrete analogue of the inner normal derivative, see [18,31].…”

“…There is the Catalan number C k of possible patterns of interfaces matching these marked points and only 2 k−1 ≪ C k correlations of disorders on free arcs to handle. The situation with domain walls is even worse: no geometric information on those can be extracted directly in discrete: already in the simplest possible setup with four marked points and '+/ − / + /−' boundary conditions, the only available way to prove the convergence of the crossing probabilities is first to prove the convergence of interfaces to hypergeometric SLEs and then to do computations in continuum, see [31].…”

Lectures on the Ising and Potts Models on the Hypercubic Lattice

“…◻ Let us conclude this section by recalling that crossing probabilities in rectangles are expected to converge to explicit functions of ρ as n tends to infinity. More generally, crossing probabilities in topological rectangles should be conformally invariant; see [113] for the case of site percolation (see also [14,121] for reviews) and [37,16,85] for the case of the Ising model.…”

Toward a conformal field theory for Schramm-Loewner evolutions

“…One way to mathematically formulate such a connection is in terms of random geometry, where topological or geometric properties of the models are associated to conformally invariant objects. Recently, this approach has been very successful for two-dimensional systems: examples include the conformal invariance of crossing probabilities in critical models [Car92,LPSA94,LLSA00,Smi01], their relationship with correlation functions in conformal field theory (see [BBK05,Izy15,FSKZ17,PW18], and references therein), the description of critical planar interfaces in terms of conformally invariant random curves (Schramm-Loewner evolutions) [Sch00, Smi01, LSW04, Smi06, SS09, SS13, CDCH + 14], and a random geometry formulation of 2D quantum gravity [Pol81,Dup04,Gal13,Mie13,DMS14,MS16a]. In this article, we focus on the relationship of Schramm-Loewner evolutions with correlation functions in conformal field theory.…”