Convergent series for lattice models with polynomial interactions

The standard perturbative weak-coupling expansions in lattice models are asymptotic. The reason for this is hidden in the incorrect interchange of the summation and integration. However, substituting the Gaussian initial approximation of the perturbative expansions by a certain interacting model or regularizing original lattice integrals, one can construct desired convergent series. In this paper we develop methods, which are based on the joint and separate utilization of the regularization and new initial approximation. We prove, that the convergent series exist and can be expressed as re-summed standard perturbation theory for any model on the finite lattice with the polynomial interaction of even degree. We discuss properties of such series and study their applicability to practical computations on the example of the lattice ${\varphi }^4$-model. We calculate $〈{\varphi }_n^2〉$ expectation value using the convergent series, the comparison of the results with the Borel re-summation and Monte Carlo simulations shows a good agreement between all these methods.