On multiple SLE for the FK-Ising model

Abstract

We prove convergence of multiple interfaces in the critical planar q = 2 random cluster model, and provide an explicit description of the scaling limit. Remarkably, the expression for the partition function of the resulting multiple SLE 16/3 coincides with the bulk spin correlation in the critical Ising model in the half-plane, after formally replacing a position of each spin and its complex conjugate with a pair of points on the real line. As a corollary, we recover Belavin-Polyakov-Zamolodchikov equations for the spin correlations.

Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in $c=-2$ Logarithmic CFT

“…We recommend readers only interested in the scaling limit results per se to first glance at Section 1.2 and then return to the present Section 1.1. However, we find the results in the present Section 1.1 rather interesting, as they greatly differ from the previously considered cases of the critical Ising model (κ = 3 and κ = 16/3) [PW18,Izy20], multiple LERW (κ = 2) [Kar20, KKP20,KW11], and critical percolation (κ = 6, see Appendix B to the present article). Indeed, when κ < 8, for instance the SLE κ pure partition functions can be uniquely classified as solutions to a certain PDE boundary value problem [FK15,PW19] (see also Appendix B), whereas in the present case of κ = 8, no classification is known and the techniques of [FK15,PW19] fail.…”

Multiple-SLE κ connectivity weights for rectangles, hexagons, and octagons

“…Instead, we determine A 2 by directly comparing (47) with N = 2 to the normalized rectangle connectivity weight formula (24) with (x 3 , x 4 ) → (x 2N−1 , x 2N ), previously found in section 2.2. We find that A 2 equals the m = 1 factor in (55) with the phase factor replaced by exp[−12πi/κ], so A N equals the product (55) but with the phase factor in the m = 1 factor replaced by exp[−12πi/κ]. Upon inserting this replacement into ( 47) and ( 55) with N = 2, N = 3, and N = 4 and decomposing the integration contours as we did in sections 2.2-2.4, we recover the connectivity weight formulas Π 1 (24) for the rectangle, Π 2 (32) for the hexagon, and Π 3 (39) for the octagon respectively.…”

“…where A N is a presently unspecified constant that we later choose in order to satisfy the duality condition (10) below. (Equation (55) gives the formula for A N . )…”

“…If c = 2N − 1 instead, as it does in sections 2.2-2.4, then the m = 1 factor in ( 55) must be altered. To find A N in this situation, we perform all steps spanning (49)-( 54) N − 2 times, finding that the ratio A N /A 2 equals the product of all factors in (55), excluding the factor with m = 1. Now, repeating this process one last time to determine A 2 would send the point x 2N−1 bearing the conjugate charge to x 2 , producing a complicated result.…”

“…Our results show that such expressions may reduce to fewer terms, sometimes even a single term, for small N. Further, obtaining results for rational values of κ with quantum group methods is possible by taking the limit of the quantum group results but involves many cancelling terms, and thus appears complicated. See [52][53][54][55][56][57][58][59][60][61][62][63] for these and related results from quantum group and other methods.…”