ROBUST DISCRETE COMPLEX ANALYSIS: A TOOLBOX
成果类型：
Article
署名作者：
Chelkak, Dmitry
署名单位：
Russian Academy of Sciences; Steklov Mathematical Institute of the Russian Academy of Sciences; St. Petersburg Department of the Steklov Mathematical Institute of the Russian Academy of Sciences; St. Petersburg Scientific Centre of the Russian Academy of Sciences; Saint Petersburg State University; Swiss Federal Institutes of Technology Domain; ETH Zurich
刊物名称：
ANNALS OF PROBABILITY
ISSN/ISSBN：
0091-1798
DOI：
10.1214/14-AOP985
发表日期：
2016
页码：
628-683
关键词：
摘要：
We prove a number of double-sided estimates relating discrete counterparts of several classical conformal invariants of a quadrilateral: cross-ratios, extremal lengths and random walk partition functions. The results hold true for any simply connected discrete domain Omega with four marked boundary vertices and are uniform with respect to Omega's which can be very rough, having many fiords and bottlenecks of various widths. Moreover, due to results from [Boundaries of planar graphs, via circle packings (2013) Preprint], those estimates are fulfilled for domains drawn on any infinite properly embedded planar graph Gamma subset of C (e.g., any parabolic circle packing) whose vertices have bounded degrees. This allows one to use classical methods of geometric complex analysis for discrete domains staying on the microscopic level. Applications include a discrete version of the classical Ahlfors-Beurling-Carleman estimate and some surgery technique developed for discrete quadrilaterals.