Resummation of the perturbative abundance-variance series

Derive a closed-form resummation of the infinite perturbative series for the abundance variance of fixed-point solutions to generalized Lotka–Volterra ecosystems, analogous to the resummation obtained for the mean abundance.

Background

The paper develops a diagrammatic perturbative expansion for the mean and variance of species abundances in generalized Lotka–Volterra ecosystems with random interaction matrices. The mean-abundance expansion is resummed exactly using Catalan and Narayana generating functions, but the corresponding variance expansion is evaluated only to finite order in the interaction-strength parameter.

The cavity method supplies a closed-form variance formula, but the author does not obtain an analogous all-orders resummation within the perturbative approach. The unresolved problem is therefore to find the combinatorial or diagrammatic structure needed to resum the perturbative variance series.

References

I have not been able to find a similar resummation for the abundance variance, however diagrammatic Feynman rules are expressed, following which the perturbative series can be calculated to any desirable order.

— Fixed-Point Abundances and the Onset of Extinctions in Multispecies Lotka-Volterra Ecosystems  (2610.22183 - Jacquod, 15 Oct 2026) in Abstract; Section 3, Section 6, and Section 10

I have not been able to derive combinatorial factors for each contributions as was done for $\overline N$ in Section~\ref{sec:pert_avg}.

— Fixed-Point Abundances and the Onset of Extinctions in Multispecies Lotka-Volterra Ecosystems  (2610.22183 - Jacquod, 15 Oct 2026) in Section 3, subsection “Variance of the Abundance Distribution”