The generating function for the connected correlator in the random energy model and its effective potential

Starting with two copies of the random energy model coupled with independent magnetic fields, the generating function for the connected correlator of the magnetization is exactly derived. Without use of the replica trick, it is shown that the Hessian of the generating function is symmetric under exchanging the two copies when the system is finite, but the symmetry is spontaneously broken in the low-temperature phase. It can be regarded as a rigorous realization of the replica symmetry breaking. The corresponding effective potential, which has two independent variables conjugate to the magnetic fields, is also calculated. It is singular when the two variables coincide. The singularity is consistent with that observed in the effective potentials of short-ranged disordered systems in the context of the functional renormalization group.