Discrete $Z^{\gamma}$: embedded circle patterns with the combinatorics of the square grid and discrete Painlev\'e equations

Abstract

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.