Tracy Widom law for sample covariance matrices under quadratic form concentration
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Guangchen Chen, Xing Lyu · arXiv: 2610.41416v1 · Summary page
Why cover this story this week?
A familiar law survives dependent coordinates
The largest eigenvalue of a sample covariance matrix is important in statistics because it can signal a dominant direction of variation—or an unusually large fluctuation. A new preprint establishes the Tracy–Widom law for this eigenvalue under a concentration assumption on quadratic forms. The sampled vectors remain independent of one another, but coordinates within each vector may be dependent.
That distinction broadens the reach of a distribution already central to random matrix theory. The proof establishes edge universality: after suitable centering and scaling, the extreme eigenvalue follows the same limiting law across the allowed models. In particular, the result removes an unconditionality assumption from existing work on log-concave ensembles and extends beyond that class. It strengthens the mathematical basis for treating some high-dimensional spectral fluctuations as universal without assuming independent measurements within each observation.
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