Tightness of the additional Weingarten-function bounds

Determine how tight the bounds in the Small Permutations Theorem and the n^{5/4} Theorem are.

Background

The paper establishes two quantitative large-N estimates for the unitary Weingarten function beyond the main uniform regime. The Small Permutations Theorem treats permutations whose length remains bounded as n,N tend to infinity and applies when n=o(N), while the n{5/4} Theorem gives a logarithmic estimate in a regime controlled by n{5/4}/N. The authors explicitly state that the sharpness of these two estimates is unresolved, leaving open the problem of determining whether their exponents and error terms can be improved or are optimal.

References

At the moment it is unclear how tight the bounds of Theorem \ref{Small Permutations Theorem} and Theorem \ref{n{5/4} Theorem}. This could be a further direction of research.

— Sharp Estimates for Large N Weingarten Functions  (2502.15892 - Nissim, 21 Feb 2025) in Remark following Theorem n^{5/4} Theorem, Section 1, subsection “Main Results and Discussion”