Résumé
This is a joint work with Amol Aggarwal and Patrick Lopatto. Lévy matrices are symmetric random matrices whose entry distributions lie in the domain of attraction of an α-stable law. For α<1, predictions from the physics literature suggest that high-dimensional Lévy matrices should display the following phase transition at a point Emob. Eigenvectors corresponding to eigenvalues in (−Emob,Emob) should be delocalized, while eigenvectors corresponding to eigenvalues outside of this interval should be localized. Further, Emob is given by the (presumably unique) positive solution to λ(E,α)=1, where λ is an explicit function of E and α.
We prove the following results about high-dimensional Lévy matrices.
(1) If λ(E,α)>1 then eigenvectors with eigenvalues near E are delocalized.
(2) If E is in the connected components of the set {x:λ(x,α)<1} containing ±∞, then eigenvectors with eigenvalues near E are localized.
(3) For α sufficiently near 0 or 1, there is a unique positive solution E=Emob to λ(E,α)=1, demonstrating the existence of a (unique) phase transition.
(a) If α is close to 0, then Emob scales approximately as |logα|^(−2/α).
(b) If α is close to 1, then Emob scales as (1−α)^(−1).
Our proofs proceed through an analysis of the local weak limit of a Lévy matrix, given by a certain infinite-dimensional, heavy-tailed operator on the Poisson weighted infinite tree.
