Edge effects in some perturbations of the Gaussian unitary ensemble

Bassler, Kevin E.; Forrester, Peter J.; Frankel, N. E.

doi:10.1063/1.3521288

Cited by 7 publications

(6 citation statements)

References 20 publications

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“…Notice from Fig. 2 that, in the case shown, pairs of eigenvalues split-off or separate [32,33] from the bulk continuous distribution at the small end of the spectrum. These separated eigenvalues include the smallest nonzero one.…”

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confidence: 80%

Mesoscopic structures and the Laplacian spectra of random geometric graphs

2015

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We investigate the Laplacian spectra of random geometric graphs (RGGs). The spectra are found to consist of both a discrete and a continuous part. The discrete part is a collection of Dirac delta peaks at integer values roughly centered around the mean degree. The peaks are mainly due to the existence of mesoscopic structures that occur far more abundantly in RGGs than in non-spatial networks. The probability of certain mesoscopic structures is analytically calculated for one-dimensional RGGs and they are shown to produce integer-valued eigenvalues that comprise a significant fraction of the spectrum, even in the large network limit. A phenomenon reminiscent of Bose-Einstein condensation in the appearance of zero eigenvalues is also found.

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confidence: 80%

Mesoscopic structures and the Laplacian spectra of random geometric graphs

2015

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“…The minimization problem would then amount to studying the maximal eigenvalue of a rank-one random perturbation of GOE, which attracted a considerable interest recently, see e.g. [36,37,38,39] and whose large deviation functional is known explicitly [40]. In contrast, the problem with magnetic field is not a simple eigenvalue problem but is equivalent to a much less studied class of quadratic eigenvalue problems [2].…”

Section: Lagrange Multiplier Minimization Relation To Rmt In Perturbmentioning

confidence: 99%

Topology Trivialization and Large Deviations for the Minimum in the Simplest Random Optimization

2013

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Abstract. Finding the global minimum of a cost function given by the sum of a quadratic and a linear form in N real variables over (N − 1)− dimensional sphere is one of the simplest, yet paradigmatic problems in Optimization Theory known as the "trust region subproblem" or "constraint least square problem". When both terms in the cost function are random this amounts to studying the ground state energy of the simplest spherical spin glass in a random magnetic field. We first identify and study two distinct large-N scaling regimes in which the linear term (magnetic field) leads to a gradual topology trivialization, i.e. reduction in the total number N tot of critical (stationary) points in the cost function landscape. In the first regime N tot remains of the order N and the cost function (energy) has generically two almost degenerate minima with the Tracy-Widom (TW) statistics. In the second regime the number of critical points is of the order of unity with a finite probability for a single minimum. In that case the mean total number of extrema (minima and maxima) of the cost function is given by the Laplace transform of the TW density, and the distribution of the global minimum energy is expected to take a universal scaling form generalizing the TW law. Though the full form of that distribution is not yet known to us, one of its far tails can be inferred from the large deviation theory for the global minimum. In the rest of the paper we show how to use the replica method to obtain the probability density of the minimum energy in the large-deviation approximation by finding both the rate function and the leading pre-exponential factor. ‡ LPTENS is a Unité Propre du C.N.R.S. associéeà l'Ecole Normale Supérieure età l'Université Paris Sud arXiv:1304.0024v2 [cond-mat.dis-nn] 1 Aug 2013Topology trivialization and large deviations for the minimum in the simplest random optimization.2

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“…The analogue of the BBP theorem in the perturbed GUE setting was established by Péché (2006), Desrosiers and Forrester (2006). Bassler, Forrester and Frankel (2010) treat an interesting generalization and mention some applications to physics. We consider a simple additive rank one perturbation of the GOE obtained by shifting the mean of every entry by the same constant µ/ √ n. By orthogonal invariance, this has the same effect on the spectrum as shifting the (1,1) entry by √ n µ.…”

Section: Introductionmentioning

confidence: 98%

Limits of spiked random matrices I

Bloemendal

Virág

2012

Probab. Theory Relat. Fields

145

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Given a large, high-dimensional sample from a spiked population, the top sample covariance eigenvalue is known to exhibit a phase transition. We show that the largest eigenvalues have asymptotic distributions near the phase transition in the rank-one spiked real Wishart setting and its general beta analogue, proving a conjecture of Baik, Ben Arous and P\'ech\'e (2005). We also treat shifted mean Gaussian orthogonal and beta ensembles. Such results are entirely new in the real case; in the complex case we strengthen existing results by providing optimal scaling assumptions. One obtains the known limiting random Schr\"odinger operator on the half-line, but the boundary condition now depends on the perturbation. We derive several characterizations of the limit laws in which beta appears as a parameter, including a simple linear boundary value problem. This PDE description recovers known explicit formulas at beta=2,4, yielding in particular a new and simple proof of the Painlev\'e representations for these Tracy-Widom distributions.Comment: 34 pages; minor corrections, references update

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Edge effects in some perturbations of the Gaussian unitary ensemble

Cited by 7 publications

References 20 publications

Mesoscopic structures and the Laplacian spectra of random geometric graphs

Mesoscopic structures and the Laplacian spectra of random geometric graphs

Topology Trivialization and Large Deviations for the Minimum in the Simplest Random Optimization

Limits of spiked random matrices I

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