Polyharmonic fields and Liouville quantum gravity measures on tori of arbitrary dimension: From discrete to continuous

Abstract For an arbitrary dimension , we study: the polyharmonic Gaussian field on the discrete torus , that is the random field whose law on given by where is the Lebesgue measure and is the discrete Laplacian; the associated discrete Liouville quantum gravity (LQG) measure associated with it, that is, the random measure on where is a regularity parameter. As , we prove convergence of the fields to the polyharmonic Gaussian field on the continuous torus , as well as convergence of the random measures to the LQG measure on , for all .