Level Statistics and Localization Transitions of Lévy Matrices

Physical Review Letters
E TarquiniM Tarzia

Abstract

This work provides a thorough study of Lévy, or heavy-tailed, random matrices (LMs). By analyzing the self-consistent equation on the probability distribution of the diagonal elements of the resolvent we establish the equation determining the localization transition and obtain the phase diagram. Using arguments based on supersymmetric field theory and Dyson Brownian motion we show that the eigenvalue statistics is the same one as of the Gaussian orthogonal ensemble in the whole delocalized phase and is Poisson-like in the localized phase. Our numerics confirm these findings, valid in the limit of infinitely large LMs, but also reveal that the characteristic scale governing finite size effects diverges much faster than a power law approaching the transition and is already very large far from it. This leads to a very wide crossover region in which the system looks as if it were in a mixed phase. Our results, together with the ones obtained previously, now provide a complete theory of Lévy matrices.

References

Sep 1, 1994·Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics·P Cizeau, J P Bouchaud
Oct 7, 1991·Physical Review Letters·Y V Fyodorov, A D Mirlin
Aug 7, 2007·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Zdzisław BurdaIsmail Zahed
Jan 15, 2011·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·F L MetzD Bollé
Aug 12, 2014·Physical Review Letters·A De LucaA Scardicchio

❮ Previous
Next ❯

Citations

Oct 22, 2016·Physical Review Letters·B L AltshulerV E Kravtsov
May 6, 2017·Physical Review Letters·I García-MataG Lemarié
Jul 16, 2017·Physical Review. E·Xiangyu CaoPierre Le Doussal

❮ Previous
Next ❯

Related Concepts

Trending Feeds

COVID-19

Coronaviruses encompass a large family of viruses that cause the common cold as well as more serious diseases, such as the ongoing outbreak of coronavirus disease 2019 (COVID-19; formally known as 2019-nCoV). Coronaviruses can spread from animals to humans; symptoms include fever, cough, shortness of breath, and breathing difficulties; in more severe cases, infection can lead to death. This feed covers recent research on COVID-19.

Blastomycosis

Blastomycosis fungal infections spread through inhaling Blastomyces dermatitidis spores. Discover the latest research on blastomycosis fungal infections here.

Nuclear Pore Complex in ALS/FTD

Alterations in nucleocytoplasmic transport, controlled by the nuclear pore complex, may be involved in the pathomechanism underlying multiple neurodegenerative diseases including Amyotrophic Lateral Sclerosis and Frontotemporal Dementia. Here is the latest research on the nuclear pore complex in ALS and FTD.

Applications of Molecular Barcoding

The concept of molecular barcoding is that each original DNA or RNA molecule is attached to a unique sequence barcode. Sequence reads having different barcodes represent different original molecules, while sequence reads having the same barcode are results of PCR duplication from one original molecule. Discover the latest research on molecular barcoding here.

Chronic Fatigue Syndrome

Chronic fatigue syndrome is a disease characterized by unexplained disabling fatigue; the pathology of which is incompletely understood. Discover the latest research on chronic fatigue syndrome here.

Evolution of Pluripotency

Pluripotency refers to the ability of a cell to develop into three primary germ cell layers of the embryo. This feed focuses on the mechanisms that underlie the evolution of pluripotency. Here is the latest research.

Position Effect Variegation

Position Effect Variagation occurs when a gene is inactivated due to its positioning near heterochromatic regions within a chromosome. Discover the latest research on Position Effect Variagation here.

STING Receptor Agonists

Stimulator of IFN genes (STING) are a group of transmembrane proteins that are involved in the induction of type I interferon that is important in the innate immune response. The stimulation of STING has been an active area of research in the treatment of cancer and infectious diseases. Here is the latest research on STING receptor agonists.

Microbicide

Microbicides are products that can be applied to vaginal or rectal mucosal surfaces with the goal of preventing, or at least significantly reducing, the transmission of sexually transmitted infections. Here is the latest research on microbicides.

Related Papers

Physical Review Letters
N Lehmann, H J Sommers
Physical Review. E, Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
P Cizeau, J P Bouchaud
Physical Review. B, Condensed Matter
M Fähnle, P Braun
© 2022 Meta ULC. All rights reserved