Imbedded Circle Patterns with the Combinatorics of the Square Grid and Discrete Painlevé Equations

   Abstract. A discrete analogue of the holomorphic maps z ^γ and log(z) is studied. These maps are given by Schramm's circle pattern with the combinatorics of the square grid. It is shown that the corresponding circle patterns are imbedded and described by special separatrix solutions of discrete Painlevé equations. Global properties of these solutions, as well as of the discrete z ^γ and log(z) , are established.