[en] The renormalization group method is used in the calculation of the asymptotic of the propagator in quantum field models in the logarithmic dimension. Corrections to the well-known main approximations are computed. The four-loop contribution in the self-energy operator, obtained by us, and the results that are best-known today, namely the five-loop approximations of the β-function and of the anomalous field dimension η, are used in the renormalized four-dimensional O(N)-symmetric ${\mathrm{\varphi <!--\; ??\; -->}}^4$ theory. The triviality of this model is discussed. (paper)