This article is cited in 1 paper

Discrete $Z^{\gamma}$: Embedded Circle Patterns with the Square Grid Combinatorics and Discrete Painlevé Equations

S. I. Agafonov

Loughborough University

Abstract: We study a discrete analogue of the holomorphic map $z^{\gamma}$. It is given by Schramm's circle pattern with the square grid combinatorics. We show that the corresponding circle patterns are embedded and described by special separatrix solutions of discrete Painlevé equations. We establish global properties of these solutions and of the discrete $z^{\gamma}$.

Keywords: circle patterns, discrete conformal map, discrete Painlevé equation.

DOI: 10.4213/tmf136

 English version: Theoretical and Mathematical Physics, 2003, 134:1, 3–13

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