D-dimensional arrays of Josephson junctions, spin glasses and q-deformed harmonic oscillators

We study the statistical mechanics of a D-dimensional array of Josephson junctions in the presence of a magnetic field. In the high-temperature region, the thermodynamical properties can be computed in the limit D to infinity , where the problem is simplified; this limit is taken in the framework of the mean-field approximation. Close to the transition point, the system behaves very similarly to a particular form of spin glasses, i.e. to gauge glasses. We have noticed that in this limit, the evaluation of the coefficients of the high-temperature expansion may be mapped onto the computation of some matrix elements for the q-deformed harmonic oscillator.