Asymptotic behavior of discrete holomorphic maps z^c and log(z)

Abstract

It is shown that discrete analogs of z ^c and log(z), defined via particular “integrable” circle patterns, have the same asymptotic behavior as their smooth counterparts. These discrete maps are described in terms of special solutions of discrete Painlevé-II equations, asymptotics of these solutions providing the behaviour of discrete z ^c and log(z) at infinity.