On analyticity with respect to the replica number in random energy   models II: zeros on the complex plane
  We characterize the breaking of analyticity with respect to the replica number which occurs in random energy models via the complex zeros of the moment of the partition function. We perturbatively evaluate the zeros in the vicinity of the transition point based on an exact expression of the moment of the partition function utilizing the steepest descent method, and examine an asymptotic form of the locus of the zeros as the system size tends to infinity. The incident angle of this locus indicates that analyticity breaking is analogous to a phase transition of the second order. We also evaluate the number of zeros utilizing the argument principle of complex analysis. The actual number of zeros calculated numerically for systems of finite size agrees fairly well with the analytical results. 
Kenzo Ogure and Yoshiyuki Kabashima