Abstract
We will discuss the asymptotic behavior of the discrete analogue of the holomorphic map $z^{gamma}$ introduced in 1996 by A. I. Bobenko. The analysis, which will be outlined in the talk, is based on the relations of the subject to the theory of discrete Painlevé equations and the use of the Riemann-Hilbert method. Specifically, using the nonlinear steepest decent method of Deift and Zhou, we will show how one can prove the asymptotic formulae which was conjectured in 1999 by A. I. Bobenko and S. I. Agafonov. The emphasis will be made on the novel features in implementation of the method caused by the Fuchsian type of the underline Riemann-Hilbert problem. This talk is based on the joint work with A. I. Bobenko.