IMAGINARY GEOMETRY II: REVERSIBILITY OF SLEκ (ρ1; ρ2) FOR κ ∈ (0,4)

• 成果类型: Article
• 署名作者: Miller, Jason; Sheffield, Scott
• 署名单位: University of Cambridge; Massachusetts Institute of Technology (MIT)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP943
• 发表日期: 2016
• 页码: 1647-1722
• 关键词: erased random-walks; conformal-invariance; critical percolation; free-field; RESTRICTION; FORMULA

摘要

Given a simply connected planar domain D, distinct points x, y is an element of partial derivative D, and kappa > 0, the Schramm-Loewner evolution SLE kappa is a random continuous non-self-crossing path in (D) over bar from x to y. The SLE kappa (rho(1); rho(2)) processes, defined for rho(1), rho(2) > -2, are in some sense the most natural generalizations of SLE kappa. When kappa <= 4, we prove that the law of the time-reversal of an SLE kappa (rho(1); rho(2)) from x to y is, up to parameterization, an SLE kappa (rho(1); rho(2)) from y to x. This assumes that the force points used to define SLE kappa (rho(1); rho(2)) are immediately to the left and right of the SLE seed. A generalization to arbitrary (and arbitrarily many) force points applies whenever the path does not (or is conditioned not to) hit partial derivative D \ {x, y}. The proof of time-reversal symmetry makes use of the interpretation of SLE kappa (rho(1); rho(2)) as a ray of a random geometry associated to the Gaussian-free field. Within this framework, the time-reversal result allows us to couple two instances of the Gaussian-free field (with different boundary conditions) so that their difference is almost surely constant on either side of the path. In a fairly general sense, adding appropriate constants to the two sides of a ray reverses its orientation.