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We study the statistical mechanics of a $D$-dimensional array of Josephsonjunctions in presence of a magnetic field on a lattice of side $2$. In the hightemperature region the thermodynamical properties can be computed in the limit$D \to \infty$. A conjectural form of the thermodynamic properties in the lowtemperature phase is obtained by assuming that they are the same of anappropriate spin glass system, based on quenched disordered couplings.Numerical simulations show that this conjecture is very accurate in one regimeof the magnetic field, while it is probably slightly inaccurate in a secondregime.Comment: latex 35 pages, including figures added in uufiles forma

Replica theory and large-D Josephson junction hypercubic models