Moment Problem in Replica Method

The main objective of this paper is to discuss mathematical validity of the replica method. We focus on the observation that the replica method calculates {\\Bbb E}(logZ) from the moment sequence {\\Bbb E}(Zn)} of a positive random variable Z, or equivalently, it calculates expectation {\\Bbb E}(X) of a random variable X on the basis of its moment generating function \\overline{\\varphi}(u)={\\Bbb E}(euX) while evaluating \\overline{\\varphi}(u) only at positive integers u=1,2,…. This aspect of the replica method will be justified if the moment problem is solved affirmatively. In this paper, we provide some counterexamples for each of which the moment sequence of Z does not uniquely determine {\\Bbb E}(logZ), thereby invalidating unconditional justification of the replica method.