I have a naive question because it’s mentioned in every random matrix paper and is not explained. What does it mean to say a random matrix has localized eigenvalues? And what are some examples of it?
2026-03-25 17:35:35.1774460135
Localization of eigenvalues
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It means that with an extremely high probability you will find an eigenvalue within a particular (i.e "localized) region. In the classic case of the GOE, all the eigenvalues are almost surely found within the semi-circle, there's no way any of them will be outside unless you change the underlying distribution from which the matrix is drawn (within the unit circle for Girko's law, etc.).
On the other hand, "delocalised" would refer to cases where the eigenvalues can be found, with a finite probability, outside this band; this for instance has tremendously important implications for many physical/biological systems.