On the Convergence to the Continuum of Finite Range Lattice Covariances

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 operators with constant coefficients and fractional powers thereof. The fluctuation coefficients satisfy uniform bounds and the sequence converges in appropriate norms to a smooth, positive definite, finite range continuum function. In this note we prove that the convergence is actually exponentially fast.