Robust discrete complex analysis: A toolbox

Abstract

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.