Replica theory for Levy spin glasses**Dedicated to Professor Thomas Nattermann on the occasion of his 60th anniversary.

Abstract

Infinite-range spin-glass models with Levy-distributed interactions show a spin-glasstransition with similarities to both the Sherrington–Kirkpatrick model and to disorderedspin systems on finite connectivity random graphs. Despite the diverging moments of thecoupling distribution the transition can be analyzed within the replica approach by workingat imaginary temperature. Within the replica symmetric approximation a self-consistentequation for the distribution of local fields is derived and from the instability of theparamagnetic solution to this equation the glass transition temperature is determined. Therole of the percolation of rare strong bonds for the transition is elucidated. The resultspartly agree and partly disagree with those obtained within the cavity approach. Numericalsimulations using parallel tempering are in agreement with the transition temperaturesfound.