Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes

Bertola, Marco; Buckingham, Robert; Lee, Seung-Yeop; Pierce, Virgil U.

doi:10.48550/arxiv.1009.3894

Cited by 6 publications

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“…Next, (iv) guarantees the existence of the multiple orthogonal polynomials needed to ensure the Riemann-Hilbert problem has a solution. We note that the allowed V (z) include any convex V (z) (see the introduction of [9]).…”

Section: Assumptions and Resultsmentioning

confidence: 99%

“…We recall the setting of our work [9]. Let g(z) be the g-function associated with the orthogonal polynomials with potential V (z) (see, for instance, [23] or [25]).…”

Section: Definition Of the Critical Regimementioning

confidence: 99%

“…We now define the critical and near-crtical regimes. A more extensive context for these definition can be found in [9]. For completeness we also define the supercritical, subcritical, and jumping outlier regimes.…”

Section: Definition Of the Critical Regimementioning

confidence: 99%

“…For completeness we also define the supercritical, subcritical, and jumping outlier regimes. The supercritical and subcritical regimes are dealt with separately in [9]; we plan to consider the (non-generic)…”

Section: Definition Of the Critical Regimementioning

confidence: 99%

“…This establishes a new universality class of matrix ensembles with the local eigenvalue density near the critical point described by the r-Airy process. The universality of the supercritical and subcritical cases has been considered separately [9].…”

Section: Introductionmentioning

confidence: 99%

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Spectra of Random Hermitian Matrices with a Small-Rank External Source: The Critical and Near-Critical Regimes

et al. 2011

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Random Hermitian matrices are used to model complex systems without time-reversal invariance. Adding an external source to the model can have the effect of shifting some of the matrix eigenvalues, which corresponds to shifting some of the energy levels of the physical system. We consider the case when the n × n external source matrix has two distinct real eigenvalues: a with multiplicity r and zero with multiplicity n − r. For a Gaussian potential, it was shown by Péché [32] that when r is fixed or grows sufficiently slowly with n (a small-rank source), r eigenvalues are expected to exit the main bulk for |a| large enough. Furthermore, at the critical value of a when the outliers are at the edge of a band, the eigenvalues at the edge are described by the r-Airy kernel. We establish the universality of the r-Airy kernel for a general class of analytic potentials for r = O(n γ ) for 0 ≤ γ < 1/12.

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Section: Assumptions and Resultsmentioning

confidence: 99%

“…We recall the setting of our work [9]. Let g(z) be the g-function associated with the orthogonal polynomials with potential V (z) (see, for instance, [23] or [25]).…”

Section: Definition Of the Critical Regimementioning

confidence: 99%

Section: Definition Of the Critical Regimementioning

confidence: 99%

Section: Definition Of the Critical Regimementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spectra of Random Hermitian Matrices with a Small-Rank External Source: The Critical and Near-Critical Regimes

et al. 2011

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Rank 1 real Wishart spiked model

2012

Comm Pure Appl Math

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In this paper, we consider N -dimensional real Wishart matrices Y in the class W R (Σ, M ) in which all but one eigenvalues of Σ is 1. Let the non-trivial eigenvalue of Σ be 1 + τ , then as N , M → ∞, with M/N = γ 2 finite and non-zero, the eigenvalue distribution of Y will converge into the Marchenko-Pastur distribution inside a bulk region. When τ increases from zero, one starts to see a stray eigenvalue of Y outside of the support of the Marchenko-Pastur density. As the this stray eigenvalue leaves the bulk region, a phase transition will occur in the largest eigenvalue distribution of the Wishart matrix. In this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occur. We will first establish the results that are valid for all N and M and will use them to carry out the asymptotic analysis. In particular, we have derived a contour integral formula for the Harish-Chandra Itzykson-Zuber integral O(N ) e tr(XgY g T ) g T dg when X, Y are real symmetric and Y is a rank 1 matrix. This allows us to write down a Fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques. As a result, we obtain an integral formula for the largest eigenvalue distribution in the large N limit characterized by Painlevé transcendents. The approach used in this paper is very different from a recent paper [23], in which the largest eigenvalue distribution was obtained using stochastic operator method. In particular, the Painlevé formula for the largest eigenvalue distribution obtained in this paper is new.

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Random Matrices with Equispaced External Source

Claeys

Wang

2014

Commun. Math. Phys.

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We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends to infinity. We obtain strong asymptotics for the multiple orthogonal polynomials associated to these models, and as a consequence for the average characteristic polynomials. One feature of the multiple orthogonal polynomials analyzed in this paper is that the number of orthogonality weights of the polynomials grows with the degree. Nevertheless we are able to characterize them in terms of a pair of 2 × 1 vector-valued Riemann-Hilbert problems, and to perform an asymptotic analysis of the Riemann-Hilbert problems.

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Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes

Cited by 6 publications

References 0 publications

Spectra of Random Hermitian Matrices with a Small-Rank External Source: The Critical and Near-Critical Regimes

Spectra of Random Hermitian Matrices with a Small-Rank External Source: The Critical and Near-Critical Regimes

Rank 1 real Wishart spiked model

Random Matrices with Equispaced External Source

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