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January 2016 Robust discrete complex analysis: A toolbox
Dmitry Chelkak
Ann. Probab. 44(1): 628-683 (January 2016). DOI: 10.1214/14-AOP985


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 $\Gamma \subset\mathbb{C}$ (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.


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Dmitry Chelkak. "Robust discrete complex analysis: A toolbox." Ann. Probab. 44 (1) 628 - 683, January 2016.


Received: 1 January 2013; Revised: 1 August 2014; Published: January 2016
First available in Project Euclid: 2 February 2016

zbMATH: 1347.60050
MathSciNet: MR3456348
Digital Object Identifier: 10.1214/14-AOP985

Primary: 39A12
Secondary: 60G50

Keywords: discrete potential theory , Planar random walk

Rights: Copyright © 2016 Institute of Mathematical Statistics


Vol.44 • No. 1 • January 2016
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