  In this paper we examine isotropic Gaussian random fields defined on $\mathbb
R^N$ satisfying certain conditions. Specifically, we investigate the type of a
critical point situated within a small vicinity of another critical point, with
both points surpassing a given threshold. It is shown that the Hessian of the
random field at such a critical point is equally likely to have a positive or
negative determinant. Furthermore, as the threshold tends to infinity, almost
all the critical points above the threshold are local maxima and the saddle
points with index $N-1$. Consequently, we conclude that the closely paired
critical points above a high threshold must comprise one local maximum and one
saddle point with index $N-1$.
