The Annals of Probability

Robust discrete complex analysis: A toolbox

Dmitry Chelkak

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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.

Article information

Ann. Probab., Volume 44, Number 1 (2016), 628-683.

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 39A12: Discrete version of topics in analysis
Secondary: 60G50: Sums of independent random variables; random walks

Planar random walk discrete potential theory


Chelkak, Dmitry. Robust discrete complex analysis: A toolbox. Ann. Probab. 44 (2016), no. 1, 628--683. doi:10.1214/14-AOP985.

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