Plansherel duality between lattice field theories and its application to power-like potentials

The report will be devoted to the application of some duality between lattice theories for the needs of quantum field theory. We will start by constructing the duality itself in two ways, and then use it for the case of a scalar field with the potential $g\phi^\alpha/\alpha$ for a real $\alpha$. Next, we use these results to calculate the continuous limit of the theory with a specially renormalized coupling constant, write down some correlators and the corresponding beta function. In the end, we will talk about ways to move to a continuous limit for theories without scaling and about the possibility of using this duality to build a correspondence between strong and weak coupling regimes in various power-like theories.