Replica trick to calculate means of absolute values: applications to stochastic equations

Abstract

A version of the replica trick can be used to evaluate means of the absolute values of functions that are not positive-definite (and with some modifications the mean logarithm of the absolute value). This trick can be applied in the context of stochastic equations, where the absolute value of the Jacobian of the stochastic function in each solution must be computed. The calculation of the average number of solutions of the 'naive' mean field equations for the SK spin-glass is discussed, an extension of the considerations to other mean field equations (such as TAP) is straightforward. The BRST supersymmetry is found to provide a useful tool in uncovering some puzzling aspects in these calculations. The trick presented provides a possible solution to these problems.