Hexagonal circle patterns and integrable systems. Patterns with constant angles

The theory of circle packings and, more generally, of circle patterns enjoys in recent years a fast development and a growing interest of specialists in complex analysis and discrete mathematics. This interest was initiated by Thurston’s rediscovery of the Koebe-Andreev theorem [K] about circle packing realizations of cell complexes of a prescribed combinatorics and by his idea about approximating the Riemann mapping by circle packings (see [T1, RS]). Since then many other remarkable facts about circle patterns were established, such as the discrete maximum principle and Schwarz’s lemma [R] and the discrete uniformization theorem [BS]. These and other results demonstrate surprisingly close analogy to the classical theory and allow one to talk about an emerging of the ”discrete analytic function theory” [DS], containing the classical theory of analytic functions as a small circles limit. Approximation problems naturally lead to infinite circle patterns for an analytic description of which it is advantageous to stick with fixed regular combinatorics. The most popular are hexagonal packings where each circle touches exactly six neighbors. The C∞ convergence of these packings to the Riemann mapping was established in [HS]. Another interesting and elaborated class with similar approximation properties to be mentioned here are circle patterns with the combinatorics of the square grid introduced by Schramm [S]. The square grid combinatorics of Schramm’s patterns results in an analytic description which is closer to the Cauchy-Riemann equations of complex analysis then the one of the packings with hexagonal combinatorics. Various other regular combinatorics also have similar properties [H]. Although computer experiments give convincing evidence for the existence of circle packing analogs of many standard holomorphic functions [DS], the only circle packings that have been described explicitly are Doyle spirals [BDS] (which are analogs of the exponential function) and conformally symmetric packings [BH] (which are analogs of a quotient of Airy functions). Schramm’s patterns are richer with explicit examples: discrete analogs of the functions exp(z), erf(z), Airy [S] and zc, log(z) [AB] are known. Moreover erf(z) is also an entire circle pattern.3 A natural question is: what property is responsible for this comparative richness of Schramm’s patterns? Is it due to the packing pattern or (hexagonal square) combinatorics difference? Or maybe it is the integrability of Schramm’s patterns which is crucial. Indeed, Schramm’s square grid circle patterns in conformal setting are known to be described by an integrable system [BP2] whereas for the packings it is still unknown4. E–mail: bobenko @ math.tu-berlin.de E–mail: timh @ sfb288.math.tu-berlin.de Doyle conjectured that the Doyle spirals are the only entire circle packings. This conjecture remains open. It should be said that, generally, the subject of discrete integrable systems on lattices different from Z is underdeveloped at present. The list of relevant publications is almost exhausted by [Ad, KN, ND].