On the convergence to the continuum of finite range lattice covariances (Q438761)

The authors consider rescaled fluctuation covariances (the existence of which was proved by \textit{D. C. Brydges, G. Guadagni} and \textit{P. K. Mitter} [J. Stat. Phys. 115, No. 1--2, 415--449 (2004; Zbl 1157.82304)]) on the sequence of lattices \((L^{-n}\mathbb{Z})^d\subset\mathbb{R}^d\) which are nested, \((L^{-n}\mathbb{Z})^d \subset (L^{-(n+1)}\mathbb{Z})^d\), and prove that this sequence converges in appropriate norms at the rate \(L^{-n/2}\) to a smooth, positive definite, finite range continuum function.