Discrete $Z^γ$: embedded circle patterns with the combinatorics of the square grid and discrete Painlevé equations

Details Discrete $Z^γ$: embedded circle patterns with the combinatorics of the square grid and discrete Painlevé equations Discrete $Z^γ$: embedded circle patterns with the combinatorics of the square grid and discrete Painlevé equations

2001-10-17

arxiv.org/abs/nlin/0110030v2

Theor. and Math. Physics, V 134, N 1, 2003, pp. 3--13

Nonlinear Sciences

Exactly Solvable and Integrable Systems

12 pagees 4 figures

Scientific paper

A discrete analog of the holomorphic map $z^{\gamma}$ is studied. It is given by Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding circle patterns are embedded and described by special separatrix solutions of discrete Painlev\'e equations. Global properties of these solutions, as well as of the discrete $z^{\gamma}$, are established.

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