Exact Analytic Continuation with Respect to the Replica Number in the Discrete Random Energy Model of Finite System Size

An expression for the moment of partition function valid for any finite system size $N$ and complex power $n(\mathrm{\Re }(n)>0)$ is obtained for a simple spin glass model termed the {\em discrete random energy model} (DREM). We investigate the behavior of the moment in the thermodynamic limit $N \to \infty$ using this expression, and find that a phase transition occurs at a certain real replica number when the temperature is sufficiently low, directly clarifying the scenario of replica symmetry breaking of DREM in the replica number space {\em without using the replica trick}. The validity of the expression is numerically confirmed.