One more discussion of the replica trick: the example of the exact solution

Abstract

A systematic replica field theory calculations are analysed using the examples of two particular one-dimensional "toy" random models with Gaussian disorder. Due to apparent simplicity of the model the replica trick calculations can be followed here step by step from the very beginning till the very end. In this way it can be easily demonstrated that formally at certain stage of the calculations the implementation of the standard replica program is just impossible. On the other hand, following the usual "doublethink" traditions of the replica calculations (i.e. closing eyes on the fact that certain suggestions used in the calculations contradict to each other) one can easily fulfil the programme till the very end to obtain physically sensible result for the entire free energy distribution function.

Universal randomness

“…The problem is that, first, such analytic continuation is not always unambiguous (see e.g. [33,34]), and second, any approximations in the calculations of the integer-valued replica partition function are quite risky for the validity of the further analytic continuation to non-integer values of N .…”

“…In order to do so one has to computeZ[N, L] for an arbitrary integer N and then perform analytical continuation of this function from integer to arbitrary complex values of N . This is the standard strategy of the replica method in disordered systems where it is well known that very often the procedure of such analytic continuation turns out to be rather controversial point [32,33]. Even in rare cases when the derivation of the replica partition function Z(N ) = Z N can be done exactly, its further analytic continuation to non-integer N appears to be ambiguous.…”

Statistical Properties of One-Dimensional Directed Polymers in a Random Potential

“…x a x b Ψ(x|y; t) (9.13) with the initial condition Ψ N x|y; t = 0 = N a=1 δ(y a − x a ). One can easily check that this solution is [76]: Note that the above expression for the wave function eq. (9.14) is valid only at finite time interval: t < π 2 (βN u) −1/2 ≡ t c (N ).…”

“…In order to do so one has to compute Z(N ; t) for an arbitrary integer N and then perform analytical continuation of this function from integer to arbitrary complex values of N . This is the standard strategy of the replica method in disordered systems where it is well known that very often the procedure of such analytic continuation turns out to be rather controversial point [71,76]. Even in rare cases where the replica partition function Z(N ; t) can be derived exactly, its further analytic continuation to non-integer N appears to be ambiguous.…”

“…In the present model this distribution can be computed explicitly and the resulting function P x|y;t (F ) turns out to be rather non-trivial [76,116,117]. Mapping the considered system to quantum bosons (see Chapter III) one introduces the N -particle wave function:…”