ANALYTIC DETERMINATION OF THE COMPLEX FIELD ZEROS OF REM

The complex field zeros of the Random Energy Model are analytically determined. For T < T(c) they are distributed in the whole complex plane with a density that decays very fast with the real component of H. For T > T(c) a region is found which is free of zeros and is associated to the paramagnetic phase of the system. This region is separated from the former by a line of zeros. The real axis is found to be enclosed in a cloud of zeros in the whole frozen phase of the system.