Limits of conformal images and conformal images of limits for planar random curves. (arXiv:1810.05608v1 [math-ph])

Consider a chordal random curve model on a planar graph, in the scaling limit when a fine-mesh graph approximates a simply-connected planar domain. The well-known precompactness conditions of Kemppainen and Smirnov show that certain "crossing estimates" guarantee the subsequential weak convergence of the random curves in the topology of unparametrized curves, as well as in a topology inherited from curves on the unit disc via conformal maps. We complement this result by proving that proceeding to weak limit commutes with changing topology, i.e., limits of conformal images are conformal images of limits, without imposing any boundary regularity assumptions on the domains where the random curves lie. Treating such rough boundaries becomes necessary, e.g., in convergence proofs to multiple SLEs. The result in this generality has not been explicated before and is not trivial, which we demonstrate by giving warning examples and deducing strong consequences. 查看全文>>