The rigidity of Doyle circle packings on the infinite hexagonal triangulation

Peter Doyle conjectured that locally univalent circle packings on the hexagonal lattice only consist of regular hexagonal packings and Doyle spirals, which is called the Doyle conjecture. In this paper, we prove a rigidity theorem for Doyle spirals in the class of infinite circle packings on the hexagonal lattice whose radii ratios of adjacent circles have a uniform bound. This gives a partial answer to the Doyle conjecture. Based on a new observation that the logarithmic of the radii ratio of adjacent circles is a weighted discrete harmonic function, we prove the result via the Liouville theorem of discrete harmonic functions.