Abstract
We construct martingale observables for systems of multiple SLE curves by applying screening techniques within the CFT framework recently developed by Kang and Makarov, extended to admit multiple SLEs. We illustrate this approach by rigorously establishing explicit expressions for the Green’s function and Schramm’s formula in the case of two curves growing towards infinity. In the special case when the curves are “fused” and start at the same point, some of the formulas we prove were predicted in the physics literature.
Highlights
Schramm–Loewner evolution (SLE) processes are universal lattice size scaling limits of interfaces in critical planar lattice models
Solving the Loewner equation gives a continuous family of conformal maps and the SLE curve is obtained by tracking the image of the singularity of the equation
To name a few examples, the SLE Green’s function, i.e., the renormalized probability that the interface passes near a given point, is important in connection with the Minkowski content parametrization [28]; Smirnov proved Cardy’s formula for the probability of a crossing event in critical percolation which entails conformal invariance [37]; left or right passage probabilities known as Schramm formulae [36] were recently used in connection with finite Loewner energy curves [39]; and observables involving derivative moments of the SLE conformal maps are important in the Communicated by Eric A
Summary
Schramm–Loewner evolution (SLE) processes are universal lattice size scaling limits of interfaces in critical planar lattice models. It was observed early on that the differential equations that arise in this way in SLE theory arise in conformal field theory (CFT) as so-called level two degeneracy equations satisfied by certain correlation functions, see [6,7,9,11,16]. Carefully chosen, field insertions, the scaling behavior at the insertion points can in some cases be prescribed In this way many chordal SLE martingale observables were recovered in [22] as CFT correlation functions. (2) The second step is to prove that the prediction from Step 1 satisfies the correct boundary conditions This technical step involves the computation of rather complicated integral asymptotics. Remark We stress that we do not need use a-priori information on the regularity of the considered observables as would be the case, e.g., if one would work directly with the differential equations
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