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International Mathematics Research Notices (2000) 2000:165-193, doi:10.1155/S1073792800000118
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Copyright © 2000 Hindawi Publishing Corporation. All rights reserved.

Discrete Z{gamma} and Painlevé equations

Sergey I. Agafonov and Alexander I. Bobenko

A discrete analogue of the holomorphic map Z{gamma} 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 Painlev' equation. Global properties of these solutions, as well as those of the discrete Z{gamma}, are established.


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