2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/96481We study the Ambegaokar-Eckern-Schön (AES) model for a regular array of metallic grains coupled by tunnel junctions of conductance $g$ and calculate both paramagnetic and diamagnetic terms in the Kubo formula for the conductivity. We find analytically, and confirm by numerical path integral Monte Carlo methods, that for $0<g<4$ the conductivity obeys an Arrhenius law $σ(T)\sim\exp[-E^{*}(g)/T]$ with an effective charging energy $E^{*} (g)$ when the temperature is sufficiently low, due to a subtle cancellation between $T^2$ inelastic-cotunneling contributions in the paramagnetic and diamagnetic terms. We present numerical results for the effective charging energy and compare the results with recent theoretical analyses. We discuss the different ways in which the experimentally observed $σ(T)\sim\exp[-\sqrt{T_{0}/T}]$ law could be attributed to disorder.5 pages, 3 figures, ReVTeX; added estimates of effective charging energies and discussion of effects of disorderMesoscale and Nanoscale PhysicsStrongly Correlated Electronstext