Exactness of the replica method in perturbation

TL;DR

This paper clarifies the mathematical validity of the replica method in perturbative field theory for disordered systems, showing that the zero-replica limit corresponds to extracting the constant term of a polynomial in the replica number.

Contribution

It provides a rigorous interpretation of the zero-replica limit as polynomial constant term extraction, validating the replica method in perturbative contexts.

Findings

Zero-replica limit corresponds to extracting the polynomial's constant term.

Mathematical clarification of the replica method's validity.

Comparison with direct calculations confirms the interpretation.

Abstract

The replica method for a quenched disordered system is considered in a perturbative field theory. Since correction in a finite-order perturbation is given in a polynomial of the replica number $n$, the zero-replica limit $n \to 0$ is regarded as extracting the constant term from the polynomial, which mathematically makes sense. The meaning of the extraction is clarified comparing with a direct calculation.