CONNECTIVITY PROPERTIES OF THE ADJACENCY GRAPH OF SLEκ BUBBLES FOR κ ∈ (4,8)

• 成果类型: Article
• 署名作者: Gwynne, Ewain; Pfeffer, Joshua
• 署名单位: University of Cambridge; Massachusetts Institute of Technology (MIT)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1402
• 发表日期: 2020
• 页码: 1495-1519
• 关键词: stable levy process; SCALING LIMITS; random-walks; Duality; dimension; cone; times; LAW

摘要

We study the adjacency graph of bubbles, that is, complementary connected components of a SLE kappa curve for kappa is an element of (4, 8), with two such bubbles considered to be adjacent if their boundaries intersect. We show that this ad- jacency graph is a.s. connected for kappa is an element of (4, kappa(0)], where kappa(0) approximate to 5.6158 is defined explicitly. This gives a partial answer to a problem posed by Duplantier, Miller and Sheffield (2014). Our proof in fact yields a stronger connectivity result for kappa is an element of (4, kappa(0)], which says that there is a Markovian way of finding a path from any fixed bubble to infinity. We also show that there is a (nonexplicit) kappa(1) is an element of (kappa(0), 8) such that this stronger condition does not hold for kappa is an element of [kappa(1), 8). Our proofs are based on an encoding of SLE kappa in terms of a pair of independent kappa/4-stable processes, which allows us to reduce our problem to a problem about stable processes. In fact, due to this encoding, our results can be rephrased as statements about the connectivity of the adjacency graph of loops when one glues together an independent pair of so-called kappa/4-stable looptrees, as studied, for example, by Curien and Kortchemski (2014). The above encoding comes from the theory of Liouville quantum gravity (LQG), but the paper can be read without any knowledge of LQG if one takes the encoding as a black box.