Fine asymptotic behavior in eigenvalues of random normal matrices: Ellipse Case
| Autor: | Seung-Yeop Lee, Roman Riser |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2015 |
| Předmět: |
15A52 (Primary)
60B20 33C45 42C05 (Secondary) FOS: Physical sciences Boundary (topology) Curvature Ellipse 01 natural sciences Normal matrix Physics::Fluid Dynamics 0103 physical sciences Classical Analysis and ODEs (math.CA) FOS: Mathematics Physics::Atomic and Molecular Clusters 0101 mathematics Eigenvalues and eigenvectors Mathematical Physics Physics Hermite polynomials Probability (math.PR) 010102 general mathematics Mathematical analysis Cauchy distribution Statistical and Nonlinear Physics Mathematical Physics (math-ph) Mathematics - Classical Analysis and ODEs Orthogonal polynomials 010307 mathematical physics Mathematics - Probability |
| Popis: | We consider the random normal matrices with quadratic external potentials where the associated orthogonal polynomials are Hermite polynomials and the limiting support (called droplet) of the eigenvalues is an ellipse. We calculate the density of the eigenvalues near the boundary of the droplet up to the second subleading corrections and express the subleading corrections in terms of the curvature of the droplet boundary. From this result we additionally get the expected number of eigenvalues outside the droplet. We also obtain the asymptotics of the kernel and found that, in the bulk, the correction term is exponentially small. This leads to the vanishing of certain Cauchy transform of the orthogonal polynomial in the bulk of the droplet up to an exponentially small error. 39 pages, 5 figures. Extended version: Theorem 1.2, Theorem 1.4, Section 6 and Section 7.3 are new |
| Databáze: | OpenAIRE |
| Externí odkaz: |
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