Abstract
We provide an exact expression for 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 Ogure and Kabashima in 2004 (Prog. Theor. Phys. 111 661). The expression can be handled either analytically or 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 using numerical methods for finite systems.