Geodesic
Discrete $Z^{\gamma}$: Embedded Circle Patterns with the Square Grid Combinatorics and Discrete Painlev\'e Equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 134 (2003) no. 1, pp. 5-17

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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 Painlevé equation.
Mots-clés : discrete conformal map 
@article{TMF_2003_134_1_a1,
     author = {S. I. Agafonov},
     title = {Discrete $Z^{\gamma}$: {Embedded} {Circle} {Patterns} with the {Square} {Grid} {Combinatorics} and {Discrete} {Painlev\'e} {Equations}},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {5--17},
     publisher = {mathdoc},
     volume = {134},
     number = {1},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2003_134_1_a1/}
}
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S. I. Agafonov. Discrete $Z^{\gamma}$: Embedded Circle Patterns with the Square Grid Combinatorics and Discrete Painlev\'e Equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 134 (2003) no. 1, pp. 5-17. http://geodesic.mathdoc.fr/item/TMF_2003_134_1_a1/