2022/10/17 by Amol Aggarwal, Aggarwal, Amol, Charles Bordenave +3 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2210.09458
openalex publication_date 2022/10/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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 |log α|-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.