Hypergeometric SLE with $\hat{I}º=8$: Convergence of UST and LERW in Topological Rectangles

Abstract

We consider uniform spanning tree (UST) in topological rectangles with alternating boundary conditions. The Peano curves associated to the UST converge weakly to hypergeometric SLE${}_8$, denoted by hSLE${}_8$. From the convergence result, we obtain the continuity and reversibility of hSLE${}_8$ as well as an interesting connection between SLE${}_8$ and hSLE${}_8$. The loop-erased random walk (LERW) branch in the UST converges weakly to SLE${}_2(-1,-1;-1,-1)$. We also obtain the limiting joint distribution of the two end points of the LERW branch.

56 pages, 6 figures. We wrap detail for the proof of tightness in Appendix C. We add Appendix D addressing hypergeometric SLE in literature