Strong freeness for one random interaction on overlapping tensor legs
Let $X_N$ be a GUE matrix of size $N^2$, normalized so that its diagonal entries have variance $N^{-2}$ and its off-diagonal entries satisfy $\mathbb{E}|(X_N)_{ab}|^2 = N^{-2}$, viewed as an operator on $\mathbb{C}^N \otimes \mathbb{C}^N$. On $(\mathbb{C}^N)^{\otimes 3}$ let $$A_N = X_N \otimes I_N, \qquad B_N = I_N \otimes X_N,$$ both built from the same realization of $X_N$. Let $s_1, s_2$ be freely independent centered semicircular elements of variance one in their reduced free product $C^*$-algebra. Collins and Yuan proved, for independent GUE matrices on overlapping groups of tensor factors under conditions on the sizes of the groups that include two matrices on legs $\{1,2\}$ and $\{2,3\}$ of $(\mathbb{C}^N)^{\otimes 3}$, that the operator norm of every noncommutative polynomial in them converges almost surely to that of the same polynomial in free semicircular elements. Chen, Garza-Vargas and van Handel proved, for independent GUE matrices acting on arbitrary sets of tensor factors, strong convergence to semicircular elements that are free when their sets of factors overlap and commute when the sets are disjoint. Collins and Yuan asked whether the same holds when one matrix is repeated on different legs, the case that arises for a quantum spin chain with one nearest-neighbour interaction.
Problem. Is it true that, for every noncommutative polynomial $P \in \mathbb{C}\langle x, y \rangle$, $$\|P(A_N, B_N)\|_{\mathrm{op}} \to \|P(s_1, s_2)\| \quad \text{in probability as } N \to \infty?$$
Source: Collins and Yuan (2026), Section 5, question (2), for the same GUE matrix acting on different legs, where they suspect freeness. Stated by AIM — Open Applied Problems in the three-site case.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- PR — Probability
- Secondary
- OA
- Keywords
- AIM Open Applied Problems, free probability, strong convergence, tensor GUE matrices
References
@article{collins2026,
author = {Collins, Beno{\^i}t and Yuan, Wangjun},
title = {Strong convergence for tensor {GUE} random matrices},
journal = {Bernoulli},
volume = {32},
number = {3},
year = {2026},
pages = {1812--1826},
doi = {10.3150/25-BEJ1933},
eprint = {2407.09065},
archivePrefix = {arXiv},
}
@article{chen2026,
author = {Chen, Chi-Fang and Garza-Vargas, Jorge and van Handel, Ramon},
title = {A new approach to strong convergence {II}. {T}he classical ensembles},
journal = {Geom. Funct. Anal.},
volume = {36},
number = {3},
year = {2026},
pages = {831--890},
doi = {10.1007/s00039-026-00744-2},
eprint = {2412.00593},
archivePrefix = {arXiv},
}
@misc{aim-repeated-gue-strong-freeness,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Strong freeness for one random interaction on overlapping tensor legs}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/577-repeated-gue-strong-freeness.md},
note = {Entry 577 at commit a153f46, 2026-10-13},
}
Resolution claims
Let $X_N$ be a normalized Gaussian unitary ensemble matrix of size $N^2$. We place the same realization on two adjacent pairs of tensor factors, as $X_N\otimes I_N$ and $I_N\otimes X_N$ on $(\C^N)^{\otimes3}$. For every fixed complex noncommutative polynomial, its operator norm converges in probability to the norm of that polynomial in two free variance-one semicircular elements. The argument computes the …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-17 | Open → Solution claimed | The registry's curators | Strong convergence for a repeated GUE interaction on three tensor legs v1 recorded from AIM @ 24edaad, paper/repeated_gue_three_legs.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00305, "Strong freeness for one random interaction on overlapping tensor legs", clopen.solutions.