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

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 $z^{\gamma}$.