THICK POINTS OF THE GAUSSIAN FREE FIELD

• 成果类型: Article
• 署名作者: Hu, Xiaoyu; Miller, Jason; Peres, Yuval
• 署名单位: Chinese Academy of Sciences; Stanford University; Microsoft
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/09-AOP498
• 发表日期: 2010
• 页码: 896-926
• 关键词: entropic repulsion; brownian-motion

摘要

Let U subset of C be a bounded domain with smooth boundary and let F be an instance of the continuum Gaussian free field on U with respect to the Dirichlet inner product integral(U) del f(x).del g(x)dx. The set T (a; U) of a-thick points of F consists of those z is an element of U such that the average of F on a disk of radius r centered at z has growth root a/pi log 1/r as r -> 0. We show that for each 0 <= a <= 2 the Hausdorff dimension of T (a; U) is almost surely 2 - a, that nu(2-a)(T (a; U)) = infinity when 0 < a <= 2 and nu(2)(T(0; U)) = nu(2)(U) almost surely, where nu(alpha) is the Hausdorff-alpha measure, and that T(a; U) is almost surely empty when a > 2. Furthermore, we prove that T(a; U) is invariant under conformal transformations in an appropriate sense. The notion of a thick point is connected to the Liouville quantum gravity measure with parameter gamma given formally by Gamma(dz) = e(root 2 pi gamma F(z)) dz considered by Duplantier and Sheffield.