Asymptotic behavior of discrete holomorphic maps z^c, log(z) and   discrete Painleve transcedents
  It is shown that discrete analogs of z^c and log(z) have the same asymptotic behavior as their smooth counterparts. These discrete maps are described in terms of special solutions of discrete Painleve-II equations, asymptotics of these solutions providing the behaviour of discrete z^c and log(z) at infinity. 
S.I.Agafonov