The scaling limit of two cluster boundaries in critical lattice models

Abstract

The probability that a point is to one side of a curve in SchrammLoewner evolution (SLEκ) can be obtained alternatively using boundary conformal field theory (BCFT). We extend the BCFT approach to treat two curves, forming, for example, the left and right boundaries of a cluster. This proves to correspond to a generalisation to SLE(κ, ρ), with ρ = 2. We derive the probabilities that a given point lies between two curves or to one side of both. We find analytic solutions for the cases κ = 0, 2, 4, 8/3, 8. The result for κ = 6 leads to predictions for the current distribution at the plateau transition in the semiclassical approximation to the quantum Hall effect.

Schramm’s Formula and the Green’s Function for Multiple SLE

“…Remark 5.7. In the fusion limit, equation (5.12) is consistent with the results of [18]. Indeed, in the limit ξ ↓ 0 the expression (5.12) for P (z, ξ) reduces to…”

“…In Section 8, we consider the special case of two fused curves, i.e., the case when both curves in the commuting system start at the same point. In the case of Schramm's formula, this provides rigorous proofs of some predictions for Schramm's formula due to Gamsa and Cardy [18].…”

“…[11]. In fact, in the case of the Schramm probability, the formulas we prove here were predicted using fusion in [18]. The formulas we derive for the fused Green's functions appear to be new.…”

Schramm’s formula for multiple loop-erased random walks

“…The corresponding equation for the denominator Φ(x) is the same except for the last two terms, which are absent. In [19] the authors noted that solving these coupled equations is already too difficult for n = 2, even when the endpoints of both curves are sent to infinity. They considered the limit of "fused" curves, namely, when the distance between the starting points of the curves goes to zero, for which they found analytical solutions of the fused BPZ equations.…”

Toward a conformal field theory for Schramm-Loewner evolutions

“…Finally, these functions admit a beautiful hierarchy of fusion rules, which can be thought of as a rigorous operator product expansion, one of the cornerstones of conformal field theory. In fact, in some cases the fusion can also be related to actual observables in critical models [GC05,KKP17] Organization of this article. We discuss the SLE, its relation to critical lattice models, and basics of CFT in Section 2.…”