Asymptotics of solutions of the discrete Painlevé I equation

Classes of asymptotic solutions of the discrete Painlevé equation of the first type (dPI) are constructed for large values of the independent variable. A special case of the semiclassical asymptotics is studied when one of the coefficients of the dPI is as large as the independent variable. An estimate is found for the transition moment at which a positive solution becomes negative. This semiclassical asymptotics generates singularities in models of Laplace growth and in distributions of eigenvalues of ensembles of normal matrices.