On the Convergence to the Continuum of Finite Range Lattice Covariances

Abstract

In (J. Stat. Phys. 115:415–449, 2004) Brydges, Guadagni and Mitter proved the existence of multiscale expansions of a class of lattice Green’s functions as sums of positive definite finite range functions (called fluctuation covariances). The lattice Green’s functions in the class considered are integral kernels of inverses of second order positive self-adjoint elliptic operators with constant coefficients and fractional powers thereof. The rescaled fluctuation covariance in the nth term of the expansion lives on a lattice with spacing L −n and satisfies uniform bounds. Our main result in this note is that the sequence of these terms converges in appropriate norms at a rate L −n/2 to a smooth, positive definite, finite range continuum function.