Robust discrete complex analysis: A toolbox

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 . with four marked boundary vertices and are uniform with respect to ..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 ..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.