Random Matrix Ensembles Associated to Compact Symmetric Spaces

Duéñez, Eduardo

doi:10.1007/s00220-003-0994-2

Cited by 24 publications

(32 citation statements)

References 23 publications

(56 reference statements)

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“…Our treatment of the derivation of joint distributions of eigenvalues was influenced by [Due04] (the latter relies directly on Weyl's formula) and [Mat97]. The book [For05] is an excellent recent reference on the derivation of joint distributions of eigenvalues of random matrices belonging to various ensembles; see also [Meh91] and the more recent [Zir96].…”

Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

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Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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“…In this paper we are concerned with the ensembles attached to compact Lie groups which are introduced by Katz-Sarnak and occur in the statistics of L-functions. (See [39] for the precise classification of these ensembles attached to different Riemannian symmetric spaces. )…”

Section: Random Matricesmentioning

confidence: 99%

Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$ L -functions

2015

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We consider certain families of automorphic representations over number fields arising from the principle of functoriality of Langlands. Let G be a reductive group over a number field F which admits discrete series representations at infinity. Let L G = G Gal(F/F) be the associated L-group and r : L G → GL(d, C) a continuous homomorphism which is irreducible and does not factor through Gal(F/F). The families under consideration consist of discrete automorphic representations of G(A F ) of given weight and level and we let either the weight or the level grow to infinity. We establish a quantitative Plancherel and a quantitative Sato-Tate equidistribution theorem for the Satake parameters of these families. This generalizes earlier results in the subject, notably of Sarnak (Prog Math 70:321-331, 1987) and Serre (J Am Math Soc 10(1):75-102, 1997). As an application we study the distribution of the lowlying zeros of the associated family of L-functions L(s, π, r ), assuming from the principle of functoriality that these L-functions are automorphic. We find that the distribution of the 1-level densities coincides with the distribution of the 1-level densities of eigenvalues of one of the unitary, symplectic and orthogonal ensembles, in accordance with the Katz-Sarnak heuristics. We provide a criterion based on the Frobenius-Schur indicator to determine this symmetry type. If r is not isomorphic to its dual r ∨ then the symmetry type is unitary. Otherwise there is a bilinear form on C d which realizes the isomorphism between r and r ∨ . If the bilinear form is symmetric (resp. alternating) then r is real (resp. quaternionic) and the symmetry type is symplectic (resp. orthogonal).

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“…In the cases β = 1 and β = 4 the limiting correlations are quaternion determinants, or equivalently Pfaffians [8,15]. This leads to a more complicated analysis than that required for the case β = 2.…”

Section: The Gap Probabilities As a Sum Over K-point Correlationsmentioning

confidence: 99%

Increasing subsequences and the hard-to-soft edge transition in matrix ensembles

Borodin¹,

Forrester²

2003

J. Phys. A: Math. Gen.

104

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Our interest is in the cumulative probabilities Pr(L(t) l) for the maximum length of increasing subsequences in Poissonized ensembles of random permutations, random fixed point free involutions and reversed random fixed point free involutions. It is shown that these probabilities are equal to the hard edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively. The gap probabilities can be written as a sum over correlations for certain determinantal point processes. From these expressions a proof can be given that the limiting form of Pr(L(t) l) in the three cases is equal to the soft edge gap probability for matrix ensembles with unitary, orthogonal and symplectic symmetry respectively, thereby reclaiming theorems due to Baik-Deift-Johansson and Baik-Rains.

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Random Matrix Ensembles Associated to Compact Symmetric Spaces

Cited by 24 publications

References 23 publications

An Introduction to Random Matrices

An Introduction to Random Matrices

Sato–Tate theorem for families and low-lying zeros of automorphic $$L$$ L -functions

Increasing subsequences and the hard-to-soft edge transition in matrix ensembles

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