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arXiv · 2506.19940

Strong convergence to operator-valued semicirculars

David Jekel·Yoonkyeong Lee·Brent Nelson·Jennifer Pi
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Abstract

We establish a framework for weak and strong convergence of matrix models to operator-valued semicircular systems parametrized by operator-valued covariance matrices $\eta = (\eta_{i,j})_{i,j \in I}$. Non-commutative polynomials are replaced by covariance polynomials that can involve iterated applications of $\eta_{i,j}$, leading to the notion of covariance laws. We give sufficient conditions for weak and strong convergence of general Gaussian random matrices and deterministic matrices to a $B$-valued semicircular family and generators of the base algebra $B$. In particular, we obtain operator-valued strong convergence for continuously weighted Gaussian Wigner matrices, such as Gaussian band matrices with a continuous cutoff, and we construct natural strongly convergent matrix models for interpolated free group factors.

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David Jekel, Yoonkyeong Lee, Brent Nelson, Jennifer Pi. 2025-06-24. Strong convergence to operator-valued semicirculars. https://arxiv.org/abs/2506.19940

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