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Weak asymptotic expansions for random permutation representations

Ascertain whether weak asymptotic expansions of polynomial spectral statistics in powers of 1/N, as in equation (5.1), hold for random permutation representation models of discrete groups, and identify which groups admit such expansions.

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Background

The polynomial method relies on the existence of weak asymptotic expansions for polynomial spectral statistics. While such expansions are known for basic random matrix models, their validity for more complex or discrete group-based models is unclear.

Determining which discrete groups admit random permutation representation models with weak asymptotic expansions would broaden the applicability of the polynomial method to new settings.

References

First, the polynomial method relies on the weak asymptotic expansion eq:weakeq. While the existence of such an expansion is an easy fact for the most basic random matrix models, it remains an open question whether or not such an expansion holds in many interesting cases. For example, it is unclear which discrete groups admit random permutation representations that have a weak asymptotic expansion.

— Strong convergence: a short survey (2510.12520 - Handel, 14 Oct 2025) in Section 5.3 (Open questions)