Weingarten asymptotics in the unstable regime

Establish whether results analogous to the paper’s large-N estimates for unitary, orthogonal, and symplectic Weingarten functions hold in the unstable regime n>N.

Background

The principal estimates in the paper concern ranges in which the matrix size N is sufficiently large relative to the number of matrix entries or pairing size n. The authors identify the complementary regime n>N as unstable and explicitly leave unresolved whether any results comparable to their theorems remain valid there. The open problem therefore concerns extending large-N Weingarten-function analysis beyond the stable parameter range addressed by the paper.

References

Another direction of interest is to study whether any result similar to the theorems of this paper hold in the unstable regime when $n>N$. We leave these questions open as a future research direction.

— 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”