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

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

doi:10.1007/s10955-011-0409-2

Cited by 17 publications

(30 citation statements)

References 46 publications

(370 reference statements)

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“…Estimation ofĨ (5) n . ForĨ (5) n andĨ (7) n , we first note that (2.20) still holds. Using a Taylor expansion and error estimate similar to the one forĨ (1) n , one verifies that (2.21) improves to…”

Section: Estimation Of I Nmentioning

confidence: 99%

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Boundaries of sine kernel universality for Gaussian perturbations of Hermitian matrices

Claeys

Neuschel

Venker

2019

Random Matrices: Theory Appl.

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We explore the boundaries of sine kernel universality for the eigenvalues of Gaussian perturbations of large deterministic Hermitian matrices. Equivalently, we study for deterministic initial data the time after which Dyson's Brownian motion exhibits sine kernel correlations. We explicitly describe this time span in terms of the limiting density and rigidity of the initial points. Our main focus lies on cases where the initial density vanishes at an interior point of the support. We show that the time to reach universality becomes larger if the density vanishes faster or if the initial points show less rigidity. e − n 2t Tr((Yn−Mn) 2 ) .

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Section: Estimation Of I Nmentioning

confidence: 99%

“…By symmetry, the integralĨ (6) n can be estimated by the same arguments, so it remains to treatĨ (5) n andĨ (7) n , which again are of the same type so we only have to deal withĨ (5) n . Estimation ofĨ (5) n .…”

Section: Estimation Ofĩmentioning

confidence: 99%

Boundaries of sine kernel universality for Gaussian perturbations of Hermitian matrices

Claeys

Neuschel

Venker

2019

Random Matrices: Theory Appl.

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“…While [8] and the current paper were being written, a work on a similar subject was announced in the recent preprints by Bertola et al [11] and [12]. The major difference of their work and ours is that we take a j 's to be all distinct and keep m fixed, while [11,12] take a j to be identical and let m → ∞ with m = o(n). Hence these two works complement each other.…”

Section: Introductionmentioning

confidence: 94%

“…The same remark applies to all other theorems in this section. It is interesting to contrast this situation to the papers [11] and [12] which analyzed the similar model using the Riemann-Hilbert problem for multiple orthogonal polynomials. In that approach, the case in which all a j 's are identical is the simplest to analyze.…”

Section: Sub-critical Casementioning

confidence: 99%

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On the Largest Eigenvalue of a Hermitian Random Matrix Model with Spiked External Source II: Higher Rank Cases

Baik

Wang

2012

International Mathematics Research Notices

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This is the second part of a study of the limiting distributions of the top eigenvalues of a Hermitian matrix model with spiked external source under a general external potential. The case when the external source is of rank one was analyzed in an earlier paper. In the present paper we extend the analysis to the higher rank case. If all the eigenvalues of the external source are less than a critical value, the largest eigenvalue converges to the right end-point of the support of the equilibrium measure as in the case when there is no external source. On the other hand, if an external source eigenvalue is larger than the critical value, then an eigenvalue is pulled off from the support of the equilibrium measure. This transition is continuous, and is universal, including the fluctuation laws, for convex potentials. For non-convex potentials, two types of discontinuous transitions are possible to occur generically. We evaluate the limiting distributions in each case for general potentials including those whose equilibrium measure have multiple intervals for their support.Here Z n is the normalization constant so that Hn p n (M )dM = 1 where dM denotes the Lebesgue measure. The sequence of probability spaces (H n , p n ), n = 1, 2, · · · , is called a Hermitian matrix model with external source matrices A n , n = 1, 2, · · · , and potential V . Note that due to the unitary invariance of dM and the presence of the trace in the exponent of (1), the density p n (M ) *

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

Cited by 17 publications

References 46 publications

Boundaries of sine kernel universality for Gaussian perturbations of Hermitian matrices

Boundaries of sine kernel universality for Gaussian perturbations of Hermitian matrices

On the Largest Eigenvalue of a Hermitian Random Matrix Model with Spiked External Source II: Higher Rank Cases

Rank 1 real Wishart spiked model

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