Extremes of homogeneous two-parametric Gaussian fields at discretization of parameters

Gaussian homogeneous fields on two-dimensional Euclidean space are considered, whose correlation functions behave at zero in a power-law manner along each of the coordinates. Exact asymptotics are evaluated for the exceedances probabilities above infinitely growing levels on lattices with different densities along each coordinates and with infinitely decreased lattice density. Relations between the evaluated asymptotic behavior and corresponding ones in continuous time at various rates of lattice densities are discussed.