Explanation of the superior cavity-method accuracy

Explain why the cavity method for generalized Lotka–Volterra abundance statistics reproduces numerical results more accurately near the boundary of the perturbative method’s validity than the controlled random-matrix perturbative approach.

Background

The paper compares two analytical approaches for the abundance mean and variance: a diagrammatic perturbative expansion based on random-matrix theory and a mean-field cavity calculation. Both agree closely over much of their common validity regime, but numerical tests near the boundary of that regime show that the cavity prediction remains accurate while the perturbative prediction begins to deviate.

This is theoretically puzzling because the cavity method relies on mean-field assumptions whose accuracy is not established a priori, whereas the perturbative method is based on apparently controlled approximations. The paper suggests that subdominant diagrams omitted by the perturbative treatment may become important near the validity boundary, but does not resolve the issue.