### Abstract

Using the theory of free random variables and the Coulomb gas analogy, we construct stable random matrix ensembles that are random matrix generalizations of the classical one-dimensional stable Levy distributions. We show that the resolvents for the corresponding matrices obey transcendental equations in the large size limit. We solve these equations in a number of cases, and show that the eigenvalue distributions exhibit Levy tails. For the analytically known Lévy measures we explicitly construct the density of states using the method of orthogonal polynomials. We show that the Levy tail distributions are characterized by a different novel form of microscopic universality.

Original language | English |
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Article number | 021106 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 65 |

Issue number | 2 |

DOIs | |

Publication status | Published - Feb 2002 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*,

*65*(2), [021106]. https://doi.org/10.1103/PhysRevE.65.021106

**Free random Lévy matrices.** / Burda, Zdzislaw; Janik, Romuald A.; Jurkiewicz, Jerzy; Nowak, Maciej A.; Papp, G.; Zahed, Ismail.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 65, no. 2, 021106. https://doi.org/10.1103/PhysRevE.65.021106

}

TY - JOUR

T1 - Free random Lévy matrices

AU - Burda, Zdzislaw

AU - Janik, Romuald A.

AU - Jurkiewicz, Jerzy

AU - Nowak, Maciej A.

AU - Papp, G.

AU - Zahed, Ismail

PY - 2002/2

Y1 - 2002/2

N2 - Using the theory of free random variables and the Coulomb gas analogy, we construct stable random matrix ensembles that are random matrix generalizations of the classical one-dimensional stable Levy distributions. We show that the resolvents for the corresponding matrices obey transcendental equations in the large size limit. We solve these equations in a number of cases, and show that the eigenvalue distributions exhibit Levy tails. For the analytically known Lévy measures we explicitly construct the density of states using the method of orthogonal polynomials. We show that the Levy tail distributions are characterized by a different novel form of microscopic universality.

AB - Using the theory of free random variables and the Coulomb gas analogy, we construct stable random matrix ensembles that are random matrix generalizations of the classical one-dimensional stable Levy distributions. We show that the resolvents for the corresponding matrices obey transcendental equations in the large size limit. We solve these equations in a number of cases, and show that the eigenvalue distributions exhibit Levy tails. For the analytically known Lévy measures we explicitly construct the density of states using the method of orthogonal polynomials. We show that the Levy tail distributions are characterized by a different novel form of microscopic universality.

UR - http://www.scopus.com/inward/record.url?scp=37649027168&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=37649027168&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.65.021106

DO - 10.1103/PhysRevE.65.021106

M3 - Article

AN - SCOPUS:37649027168

VL - 65

JO - Physical review. E

JF - Physical review. E

SN - 2470-0045

IS - 2

M1 - 021106

ER -