article published yes Oleksii Kolupaiev author 149b70d4-896a-11ed-bdf8-8c63fd44ca610000-0003-1491-4623 LaEr department Random matrices beyond Wigner-Dyson-Mehta project We study the joint spectral properties of two coupled random matrices H(1) and H(2), which are either real symmetric or complex Hermitian. The entries of these matrices exhibit polynomially decaying correlations, both within each matrix and between them. Surprisingly, we find that under extremely weak decorrelation condition, permitting H(1) and H(2) to be almost fully correlated, the fluctuations of their individual eigenvalues in the bulk of the spectrum are still asymptotically independent. Furthermore, we demonstrate that this decorrelation condition is optimal. Bernoulli Society for Mathematical Statistics and Probability2026 eng correlated random matrixeigenvalue perturbation theoryLocal lawWigner-type matrix 1350-7265 2503.2147210.3150/26-bej1976 3242999-3023 O. Kolupaiev, Bernoulli 32 (2026) 2999–3023. Kolupaiev O. Spectral independence of almost fully correlated random matrices. <i>Bernoulli</i>. 2026;32(4):2999-3023. doi:<a href="https://doi.org/10.3150/26-bej1976">10.3150/26-bej1976</a> Kolupaiev, Oleksii. “Spectral Independence of Almost Fully Correlated Random Matrices.” <i>Bernoulli</i>. Bernoulli Society for Mathematical Statistics and Probability, 2026. <a href="https://doi.org/10.3150/26-bej1976">https://doi.org/10.3150/26-bej1976</a>. Kolupaiev, Oleksii. “Spectral Independence of Almost Fully Correlated Random Matrices.” <i>Bernoulli</i>, vol. 32, no. 4, Bernoulli Society for Mathematical Statistics and Probability, 2026, pp. 2999–3023, doi:<a href="https://doi.org/10.3150/26-bej1976">10.3150/26-bej1976</a>. Kolupaiev O. 2026. Spectral independence of almost fully correlated random matrices. Bernoulli. 32(4), 2999–3023. O. Kolupaiev, “Spectral independence of almost fully correlated random matrices,” <i>Bernoulli</i>, vol. 32, no. 4. Bernoulli Society for Mathematical Statistics and Probability, pp. 2999–3023, 2026. Kolupaiev, O. (2026). Spectral independence of almost fully correlated random matrices. <i>Bernoulli</i>. Bernoulli Society for Mathematical Statistics and Probability. <a href="https://doi.org/10.3150/26-bej1976">https://doi.org/10.3150/26-bej1976</a> 231122026-10-12T12:04:45Z2026-10-14T07:34:08Z