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Apology if this question is not appropriate. I was looking to associate entropy to eigenvectors for some of my work and I found the link http://chaos.if.uj.edu.pl/~karol/pdf/ZK94.pdf . This leads to the concept of localization of eigenvectors as mentioned in https://www.researchgate.net/profile/Luca_Molinari/publication/236159789_Scaling_Properties_of_Band_Random_Matrices_Giulio_Casati_Luca_Molinari_and_Felix_Izrailev_Phys._Rev._Lett._64_1851_(1990)/links/00463516883193f770000000.pdf

The abstract of the second link is: "It is shown on the basis of numerical data that the normalised localisation length of eigenvectors of band random matrices follows a scaling law. The scaling parameteris b2/N, where Ь measures the band-width and N is the size of the matrix."

May I ask what exactly 'localization length of eigenvectors' means. I understand eigenvectors are generally unity length and it is important for direction only. Can anyone please help.

Qmechanic
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In certain disordered physical systems, the eigenstates have a localized behavior, in the sense that they fall off exponentially in space like $\psi(x) \approx e^{-x/\xi}$, with $\xi$ defined as the localization length. This is called Anderson localization. If you search for that term here, you can find more information about it, see for example here or here.

I think in this paper they are trying to talk about general properties of the matrices that describe such systems, from a mathematical perspective, so they talk about localization lengths of eigenvectors instead of eigenstates. Out of context, this is indeed rather confusing, but if you reread their introduction with that mindset maybe it will make more sense.

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Rococo
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The author has extended the idea of Anderson localization as mentioned in the wikipedia link :https://en.wikipedia.org/wiki/Anderson_localization The Anderson localization is defined on wave function (probability distribution). the eigenvectors are associated with wave function and so the connection and hence the author termed it.

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