Hypergeometric SLE with $\kappa=8$: Convergence of UST and LERW in Topological Rectangles

Yong Han; Mingchang Liu; Hao Wu

Hypergeometric SLE with $\kappa=8$: Convergence of UST and LERW in Topological Rectangles

Probability 2023-10-24 v5

Authors: Yong Han , Mingchang Liu , Hao Wu

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.

Keywords

uniform spanning tree loop-erased random walk hypergeometric sle topological rectangle weak convergence peano curve sle8

Cite

@article{arxiv.2008.00403,
  title  = {Hypergeometric SLE with $\kappa=8$: Convergence of UST and LERW in Topological Rectangles},
  author = {Yong Han and Mingchang Liu and Hao Wu},
  journal= {arXiv preprint arXiv:2008.00403},
  year   = {2023}
}

Comments

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

Hypergeometric SLE with $\kappa=8$: Convergence of UST and LERW in Topological Rectangles

Abstract

Keywords

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