LOEWNER EVOLUTION DRIVEN BY COMPLEX BROWNIAN MOTION

• 成果类型: Article
• 署名作者: Gwynne, Ewain; Pfeffer, Joshua; Park, Minjae
• 署名单位: University of Chicago; Columbia University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/23-AOP1639
• 发表日期: 2023
• 页码: 2086-2130
• 关键词: erased random-walks; conformal-invariance; critical percolation; QUANTUM-GRAVITY; sle; Duality; dimension; FIELDS

摘要

We study the Loewner evolution whose driving function is W-t = B-t(1) + iB(t)(2), where (B-1, B-2) is a pair of Brownian motions with a given covariance matrix. This model can be thought of as a generalization of Schramm-Loewner evolution (SLE) with complex parameter values. We show that our Loewner evolutions behave very differently from ordinary SLE. For example, if neither B-1 nor B-2 is identically equal to zero, then the set of points disconnected from infinity by the Loewner hull has non-empty interior at each time. We also show that our model exhibits three phases analogous to the phases of SLE: a phase where the hulls have zero Lebesgue measure, a phase where points are swallowed but not hit by the hulls, and a phase where the hulls are space-filling. The phase boundaries are expressed in terms of the signs of explicit integrals. These boundaries have a simple closed form when the correlation of the two Brownian motions is zero.