article
Extremal statistics of quadratic forms of GOE/GUE eigenvectors
published
yes
László
Erdös
author 4DBD5372-F248-11E8-B48F-1D18A9856A870000-0001-5366-9603
Benjamin
McKenna
author b0cc634c-d549-11ee-96c8-87338c7ad8080000-0003-2625-495X
LaEr
department
Random matrices beyond Wigner-Dyson-Mehta
project
We consider quadratic forms of deterministic matrices A evaluated at the random eigenvectors of a large N×N GOE or GUE matrix, or equivalently evaluated at the columns of a Haar-orthogonal or Haar-unitary random matrix. We prove that, as long as the deterministic matrix has rank much smaller than √N, the distributions of the extrema of these quadratic forms are asymptotically the same as if the eigenvectors were independent Gaussians. This reduces the problem to Gaussian computations, which we carry out in several cases to illustrate our result, finding Gumbel or Weibull limiting distributions depending on the signature of A. Our result also naturally applies to the eigenvectors of any invariant ensemble.
Institute of Mathematical Statistics2024
eng
Annals of Applied Probability
1050-5164
2208.1220610.1214/23-AAP2000
341B1623-1662
Erdös, László, and Benjamin McKenna. “Extremal Statistics of Quadratic Forms of GOE/GUE Eigenvectors.” <i>Annals of Applied Probability</i>, vol. 34, no. 1B, Institute of Mathematical Statistics, 2024, pp. 1623–62, doi:<a href="https://doi.org/10.1214/23-AAP2000">10.1214/23-AAP2000</a>.
Erdös L, McKenna B. Extremal statistics of quadratic forms of GOE/GUE eigenvectors. <i>Annals of Applied Probability</i>. 2024;34(1B):1623-1662. doi:<a href="https://doi.org/10.1214/23-AAP2000">10.1214/23-AAP2000</a>
Erdös, László, and Benjamin McKenna. “Extremal Statistics of Quadratic Forms of GOE/GUE Eigenvectors.” <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2024. <a href="https://doi.org/10.1214/23-AAP2000">https://doi.org/10.1214/23-AAP2000</a>.
L. Erdös and B. McKenna, “Extremal statistics of quadratic forms of GOE/GUE eigenvectors,” <i>Annals of Applied Probability</i>, vol. 34, no. 1B. Institute of Mathematical Statistics, pp. 1623–1662, 2024.
Erdös, L., & McKenna, B. (2024). Extremal statistics of quadratic forms of GOE/GUE eigenvectors. <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/23-AAP2000">https://doi.org/10.1214/23-AAP2000</a>
L. Erdös, B. McKenna, Annals of Applied Probability 34 (2024) 1623–1662.
Erdös L, McKenna B. 2024. Extremal statistics of quadratic forms of GOE/GUE eigenvectors. Annals of Applied Probability. 34(1B), 1623–1662.
150252024-02-25T23:00:56Z2024-02-27T08:29:05Z