Solving the frustrated spherical model withq-polynomials

Abstract

We analyse the spherical model with frustration induced by an external gauge field. This has been recently mapped in infinite dimensions onto a problem of q-deformed oscillators, whose real parameter q measures the frustration. We find the analytic solution of this model by suitably representing the q-oscillator algebra with q-Hermite polynomials. We also present a related matrix model which possesses the same diagrammatic expansion in the planar approximation. Its interaction potential is oscillating at infinity with period , and may lead to interesting metastability phenomena beyond the planar approximation. The spherical model is similarly q-periodic, but does not exhibit such phenomena: actually its low-temperature phase is not glassy and depends smoothly on q.