On the Convergence to the Continuum of Finite Range Lattice Covariances

TL;DR

This paper proves that the convergence of lattice Green's functions to their continuum limits, previously established to be uniform and smooth, actually occurs at an exponential rate, enhancing understanding of multiscale expansions.

Contribution

It demonstrates that the convergence of fluctuation covariances to continuum functions is exponentially fast, improving prior results on their uniform bounds and limits.

Findings

Convergence of lattice Green's functions to continuum functions is exponential.

Uniform bounds on fluctuation covariances are maintained during convergence.

The results apply to inverses of second order positive self-adjoint operators with fractional powers.

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 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.