  The main aim of this work - to calculate 2- and 4-point connected
contributions to the $S$-matrix and correlation functions for the euclidean
scalar field on the lattice with self-action $|\phi|^{n}$ for $n>2$ with
coupling constant renormalised in a special way for arbitrary dimension. It is
shown that the considered theory has a nontrivial continuous limit. A new
method is proposed without the use of perturbation theory and diagrams. We have
used the explored duality between different lattice field theories. After its
application, it turns out that it is possible to apply the saddle-point method
to the generating functional, where the concentration of nodes of the initial
lattice approximation acts as a large parameter. In addition to the connected
contributions to the $S$-matrix, the beta function is calculated. We have found
a critical point for the theory of $\phi^{4k}$ in arbitrary dimension and a
nontrivial mass gap of the interacting massless theory. We have received the
masses of bound and single-particle states of the interacting theory. The
proposed method can be extend for application to QED and gauge theories, as
well as generalized to the arbitrary geometry of space. Though the way of
coupling rescaling is specific, it gives nontrivial theory with the majority of
perturbatively expected effects.
