Imaginary geometry III: reversibility of SLEk for k ∈ (4,8)

• 成果类型: Article
• 署名作者: Miller, Jason; Sheffield, Scott
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2016.184.2.3
• 发表日期: 2016
• 页码: 455-486
• 关键词: gaussian free-field; conformal loop ensembles; uniform spanning-trees; erased random-walks; critical percolation; SCALING LIMITS; ising-model; Invariance; CONSTRUCTION; Couplings

摘要

Suppose that D subset of C is a Jordan domain and x, y is an element of partial derivative D are distinct. Fix k is an element of (4, 8), and let eta be an SLEk process from x to y in D. We prove that the law of the time-reversal of eta is, up to reparametrization, an SLEk, process from y to x in D. More generally, we prove that SLEk(rho(1); rho(2)) processes are reversible if and only if both rho(i), are at least k/2 - 4, which is the critical threshold at or below which such curves are boundary filling. Our result supplies the missing ingredient needed to show that for all k is an element of (4, 8), the so-called conformal loop ensembles CLE kappa, are canonically defined, with almost surely continuous loops. It also provides an interesting way to couple two Gaussian free fields (with different boundary conditions) so that their difference is piecewise constant and the boundaries between the constant regions are SLEk, curves.