Differential Operators on a Semisimple Lie Algebra

Harish-Chandra,

doi:10.2307/2372387

Cited by 473 publications

(447 citation statements)

References 12 publications

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“…It happens that, for the deformed Wigner ensemble, the computation of the asymptotics of correlation functions is quite simple. This follows from beautiful results of [9], [3], [8], [7].…”

Section: Sketch Of the Proofmentioning

confidence: 84%

The largest eigenvalue of small rank perturbations of Hermitian random matrices

Péché

2005

Probab. Theory Relat. Fields

184

229

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We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. We consider random Hermitian matrices with independent Gaussian entries M ij , i ≤ j with various expectations. We prove that the largest eigenvalue of such random matrices exhibits, in the large N limit, various limiting distributions depending on both the eigenvalues of the matrix (EM ij ) N i,j=1and its rank. This rank is also allowed to increase with N in some restricted way.

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“…It happens that, for the deformed Wigner ensemble, the computation of the asymptotics of correlation functions is quite simple. This follows from beautiful results of [9], [3], [8], [7].…”

Section: Sketch Of the Proofmentioning

confidence: 84%

The largest eigenvalue of small rank perturbations of Hermitian random matrices

Péché

2005

Probab. Theory Relat. Fields

184

229

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“…. , a N ) can be computed using the Harish-Chandra formula [HC57a,HC57b] (sometimes known also as Itzykson-Zuber [IZ80] formula in physics literature):…”

Section: Gafa Lie Groups and Quantized Free Convolution 767mentioning

confidence: 99%

Representations of classical Lie groups and quantized free convolution

Bufetov

Gorin

2015

Geom. Funct. Anal.

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Abstract. We study the decompositions into irreducible components of tensor products and restrictions of irreducible representations for all series of classical Lie groups as the rank of the group goes to infinity. We prove the Law of Large Numbers for the random counting measures describing the decomposition. This leads to two operations on measures which are deformations of the notions of the free convolution and the free projection. We further prove that if one replaces counting measures with others coming from the work of Perelomov and Popov on the higher order Casimir operators for classical groups, then the operations on the measures turn into the free convolution and projection themselves. We also explain the relation between our results and limit shape theorems for uniformly random lozenge tilings with and without axial symmetry.

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“…known as the Harish-Chandra-Itzykson-Zuber formula [50,55], see also [46]. Here the eigenvalues of Σ −1 are given by diag(π j ) and V (l) is the Vandermonde determinant (13).…”

Section: Beyond the Null Hypothesismentioning

confidence: 99%

High dimensional statistical inference and random matrices

Johnstone¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

132

174

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Abstract. Multivariate statistical analysis is concerned with observations on several variables which are thought to possess some degree of inter-dependence. Driven by problems in genetics and the social sciences, it first flowered in the earlier half of the last century. Subsequently, random matrix theory (RMT) developed, initially within physics, and more recently widely in mathematics. While some of the central objects of study in RMT are identical to those of multivariate statistics, statistical theory was slow to exploit the connection. However, with vast data collection ever more common, data sets now often have as many or more variables than the number of individuals observed. In such contexts, the techniques and results of RMT have much to offer multivariate statistics. The paper reviews some of the progress to date. Mathematics Subject Classification (2000). Primary 62H10; 62H25; 62H20; Secondary 15A52.

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Differential Operators on a Semisimple Lie Algebra

Cited by 473 publications

References 12 publications

The largest eigenvalue of small rank perturbations of Hermitian random matrices

The largest eigenvalue of small rank perturbations of Hermitian random matrices

Representations of classical Lie groups and quantized free convolution

High dimensional statistical inference and random matrices

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