Replica mean-field theory for Levy spin-glasses

Infinite-range spin-glass models with Levy-distributed interactions show a freezing transition similar to disordered spin systems on finite connectivity random graphs. It is shown that despite diverging moments of the local field distribution this transition can be analyzed within the replica approach by working at imaginary temperature and using a variant of the replica method developed for diluted systems and optimization problems. The replica-symmetric self-consistent equation for the distribution of local fields illustrates how the long tail in the distribution of coupling strengths gives rise to a significant fraction of strong bonds per spin which form a percolating backbone at the transition temperature.