Moment methods on compact groups Weingarten calculus and its applications · 2022-ana-190

Source-reported assessment from the ICM catalogue snapshot. This is not an independent review by Frontier Science.

Verdict: unresolved.

The normalized-trace part of the problem is known. By strong asymptotic freeness of independent Haar unitary matrices, every normalized trace of a word in $U_1^{(n)},\ldots,U_d^{(n)}$ converges almost surely to the corresponding free-group trace. A monomial in the tensors $U_i^{(n)}\otimes U_j^{(n)}$ has normalized trace equal to the product of the normalized traces of its two component words, so this gives the stated $\tau\otimes\tau$ limit for every $*$-polynomial.

The operator-norm assertion for the entire cross-indexed family is not resolved by the literature located through August 2026. Collins explicitly isolates the behavior of $U_i\otimes U_j$ for arbitrary indices, rather than only $i=j$, as an open direction. Subsequent results prove strong convergence for diagonal tensor representations and for models whose different tensor legs use independent Haar families, but neither result covers the dependence in this problem, where the same tuple $(U_1^{(n)},\ldots,U_d^{(n)})$ is reused on both legs. No proof or counterexample for that exact joint norm convergence was found.

Key references:

• Benoit Collins, Moment methods on compact groups: Weingarten calculus and its applications, Proceedings of the International Congress of Mathematicians 2022, Vol. 4, 2022, pp. 3142-3164, DOI 10.4171/ICM2022/114. Section 6(4) identifies arbitrary-index tensors $U_i\otimes U_j$ as the outstanding case and contrasts it with the diagonal case treated in earlier work.
• Benoit Collins and Camille Male, The strong asymptotic freeness of Haar and deterministic matrices, Annales scientifiques de l'Ecole Normale Superieure 47 (2014), no. 1, 147-163, DOI 10.24033/asens.2211. Their strong convergence theorem for independent Haar unitaries supplies, in particular, the almost-sure word-trace limits used above; it does not control the correlated tensor-square model.
• Charles Bordenave and Benoit Collins, Strong asymptotic freeness for independent uniform variables on compact groups associated to nontrivial representations, Inventiones Mathematicae 237 (2024), no. 1, 221-273, DOI 10.1007/s00222-024-01259-z. Their tensor theorem covers tuples such as $(U_i^{\otimes q})_i$, hence diagonal index patterns, but not the joint family $(U_i\otimes U_j)_{i,j}$.
• Charles Bordenave and Benoit Collins, Norm of matrix-valued polynomials in random unitaries and permutations, preprint, 2024 version, arXiv:2304.05714. Theorem 9.2 proves strong convergence in probability for tensor-product representations of Cartesian products of free groups when the Haar matrices on distinct tensor legs are independent. Its independence hypothesis excludes the reused-family model here, even though it is the strongest closely related tensor result found.