SLE duality

Date: 2015-10-07; view: 388.

For κ > 4 the curve continually touches itself and therefore its hull K_t contains earlier portions of the trace (see Fig. 4). However, the frontier of K_t (i.e., the boundary of H K_t, minus any portions of the real axis), is by definition a simple curve. A beautiful result, first suggested by Duplantier [20], and proved by Beffara [21] for the case κ = 6, is that locally this curve is an , with

(21)

For example, the exterior of a percolation cluster contains many ‘fjords' which, on the lattice, are connected to the main ocean by a neck of water which is only a finite number of lattice spacings wide. These are sufficiently frequent and the fjords macroscopically large that they survive in the continuum limit. SLE₆ describes the boundaries of the clusters, including the coastline of all the fjords. However, the coastline as seen from the ocean is a simple curve, which is locally SLE_8/3, the same as that conjectured for a self-avoiding walk. This suggests, for example, that locally the frontier of a percolation cluster and a self-avoiding walk are the same in the scaling limit. In Section 5, we show that SLE_κ and correspond to CFTs with the same value of the central charge c.

3.5. Special values of κ