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

For an arbitrary dimension n, we study: the polyharmonic Gaussian field hL on the discrete torus T-L(n)=1/L Z(n)/Z(n), that is the random field whose law on R-TLn given by c(n)e(-bn)& Vert;(-Delta L)(n/4h)& Vert;(2)dh, where dh is the Lebesgue measure and Delta(L) is the discrete Laplacian; the associated discrete Liouville quantum gravity (LQG) measure associated with it, that is, the random measure on T-L(n) mu(L)(dz)=exp(gamma h(L)(z)-gamma(2)/ 2 Eh(L)(z))dz, where gamma is a regularity parameter. As L ->infinity, we prove convergence of the fields h(L) to the polyharmonic Gaussian field h on the continuous torus T-n=R-n/Z(n), as well as convergence of the random measures mu(L) to the LQG measure mu on Tn, for all |gamma|< root 2n.