Discrete Z^a and Painleve equations

Sergey I. Agafonov; Alexander I. Bobenko

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Discrete Z^a and Painleve equations

Sergey I. Agafonov·Alexander I. Bobenko

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Abstract

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 Painleve equation. Global properties of these solutions, as well as of the discrete $z^a$ are established.

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Sergey I. Agafonov, Alexander I. Bobenko. 1999-09-01. Discrete Z^a and Painleve equations. https://arxiv.org/abs/solv-int/9909002

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