Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 404

Discrete $Z^gamma$ and Painlevé equations

Sergey I. Agafonov and Alexander I. Bobenko

A discrete analogue of the holomorphic map $z^a$ is studied. It is given by a Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding immersed circle patterns lead to special separatrix solutions of a discrete Painlevé equation. Global properties of these solutions, as well as of the discrete $z^a$ are established.

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