  We provide an exact expression of the moment of the partition function for
random energy models of finite system size, generalizing an earlier expression
for a grand canonical version of the discrete random energy model presented by
the authors in Prog. Theor. Phys. 111, 661 (2004). The expression can be
handled both analytically and numerically, which is useful for examining how
the analyticity of the moment with respect to the replica numbers, which play
the role of powers of the moment, can be broken in the thermodynamic limit. A
comparison with a replica method analysis indicates that the analyticity
breaking can be regarded as the origin of the one-step replica symmetry
breaking. The validity of the expression is also confirmed by numerical methods
for finite systems.
