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

Abstract

Systematic replica field theory calculations are analysed using the examples of two particular one-dimensional ‘toy’ random models with Gaussian disorder. Due to the simplicity of the models an integer n-th power of the partition function can be calculated here exactly. However, further analytic continuation for non-integer values of the replica parameter n inevitably involves the usual replica method of ‘cheating’, which nevertheless allows us to derive correct and rather non-trivial results for the entire free energy distribution functions both for a finite system size and in the thermodynamic limit.