Critical Ising interfaces in multiply-connected domains

Abstract

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]

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

“…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 .…”

“…In Section 3, we derive the convergence of the interfaces. This part is quite standard and employs the same argument as e. g. in [Izy13,Zha08], eventually going back to [LSW04]. In Section 4, we give the explicit formulae for the observables and hence for the drift terms.…”

“…We stick for simplicity to the case of simply connected domains with arbitrary number of boundary arcs, carrying +, −, or free boundary conditions, although in principle, using techniques employed in [Izy13] for fixed boundary conditions, one can extend these results to multiply connected domains. Our result (Theorem 3.1, see Section 3 for details and notation) states that in the limit, the interfaces are conformally invariant and are described by the chordal Loewner evolution with a driving force a 1 (t) satisfying the SDE da 1 (t) = √ 3dB t + D({a i (t), b i (t)})dt, where a i (t) and b i (t) are the images under the Loewner map at time t of the points where boundary conditions change (from + to − and from fixed to free, respectively) and D is a quadratic irrational drift function.…”

Configurations of FK Ising interfaces and hypergeometric SLE

“…It was shown in [14], that the interface of this model in the chordal setup converges to SLE (6). Note that the existence of an open percolation crossing from [a δ b δ ] to [c δ d δ ] in Ω δ is exactly the event of an internal arc pattern (a δ d δ , c δ b δ ) of interfaces.…”

Schramm’s formula for multiple loop-erased random walks

“…25) where deg s (S) and deg s (Γ) are the products of the degrees (on G s ) of all vertices belonging to S and Γ, respectively. Hence we find…”