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In J. Stat. Phys. 115, 415-449 (2004) Brydges, Guadagni and Mitter proved theexistence of multiscale expansions of a class of lattice Green's functions assums of positive definite finite range functions (called fluctuationcovariances). The lattice Green's functions in the class considered areintegral kernels of inverses of second order positive self adjoint operatorswith constant coefficients and fractional powers thereof. The fluctuationcoefficients satisfy uniform bounds and the sequence converges in appropriatenorms to a smooth, positive definite, finite range continuum function. In thisnote we prove that the convergence is actually exponentially fast.

On the Convergence to the Continuum of Finite Range Lattice Covariances