True decay rate of the generalized symmetric-group largest-part distribution

Establish the true decay rate, for large m, of the distribution function $\psi_1(u)$ for the largest part of the self-similar partition generated by the generalized symmetric group $C_m^n \rtimes S_n$, and prove the conjecture that this decay is substantially faster than the upper bound derived in the paper.

Background

For the generalized symmetric group, the limiting partition has equal-sized base components whose size is determined by divisors l of m with probabilities φ(l)/m. The corresponding largest-part distribution function is denoted by ψ_1(u).

The paper derives an iterated upper bound for ψ_1(u) by replacing a weighted historical integral with a coarse rectangular estimate. The authors explain that this estimate becomes increasingly loose as m grows because the longest look-back interval has probability only 1/m, and they explicitly conjecture that the actual decay is much faster.

References

Iterating this $\ceil{u/m}$ times gives $$ \psi_1(u) \le \frac{m{\ceil{u/m}}}{u!_{(m)}} = \frac{m{\ceil{u/m}}}{u(u-m)(u-2m)\dots} $$ but we conjecture the true decay is much more rapid for large $m$.

— Cutting a unit square and permuting blocks  (2501.13844 - Tung, 23 Jan 2025) in Section 7, Example following the discussion of identities and asymptotics