  We study the asymptotic behavior of the discrete analogue of the holomorphic
map $z^a$. The analysis is based on the use of the Riemann-Hilbert approach.
Specifically, using the Deift-Zhou nonlinear steepest descent method we prove
the asymptotic formulae which was conjectured in 2000 by the first co-author
and S.I.~Agafonov.
