Thinning and conditioning of the circular unitary ensemble

Charlier, Christophe; Claeys, Tom

doi:10.1142/s2010326317500071

Cited by 40 publications

(43 citation statements)

References 36 publications

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“…A corollary of the determinantal structure is that the gap probability

E_{N, 2} false(0; J; x^{a} e^{- x}; ξ false)

(the extra argument ξ on top of the arguments used in can be thought of as a thinning parameter: each eigenvalue is deleted independently with probability

(1 - ξ)

and therefore

0 < ξ \leq 1

(see, eg, the recent works as well as the pioneering paper), which has the operator‐theoretic form

E_{N, 2} false(0; J; x^{a} e^{- x}; ξ false) false) = trueprefixdet false(double-struckI - ξ K_{N, J} false),

where

double-struckI

is the identity operator and

K_{N, J}

is the integral operator with the kernel acting on

L^{2} false(J false)

. As a consequence, the large N expansion of for

J = (0, s / 4 N)

—referred to as a hard edge scaling—can be deduced from knowledge of the corresponding large N asymptotics of .…”

Section: Functional Form Of Optimal Correction Term For Complex Wishamentioning

confidence: 99%

Finite‐size corrections at the hard edge for the Laguerre β ensemble

Forrester

Trinh

2019

Stud Appl Math

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A fundamental question in random matrix theory is to quantify the optimal rate of convergence to universal laws. We take up this problem for the Laguerre β ensemble, characterized by the Dyson parameter β, and the Laguerre weight xae−βx/2, x>0 in the hard edge limit. The latter relates to the eigenvalues in the vicinity of the origin in the scaled variable x↦x/4N. Previous work has established the corresponding functional form of various statistical quantities—for example, the distribution of the smallest eigenvalue, provided that a∈double-struckZ≥0. We show, using the theory of multidimensional hypergeometric functions based on Jack polynomials, that with the modified hard edge scaling x↦x/4(N+a/β), the rate of convergence to the limiting distribution is O(1/N2), which is optimal. In the case β=2, general a>−1 the explicit functional form of the distribution of the smallest eigenvalue at this order can be computed, as it can for a=1 and general β>0. An iterative scheme is presented to numerically approximate the functional form for general a∈double-struckZ≥2.

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“…A corollary of the determinantal structure is that the gap probability

E_{N, 2} false(0; J; x^{a} e^{- x}; ξ false)

(the extra argument ξ on top of the arguments used in can be thought of as a thinning parameter: each eigenvalue is deleted independently with probability

(1 - ξ)

and therefore

0 < ξ \leq 1

(see, eg, the recent works as well as the pioneering paper), which has the operator‐theoretic form

E_{N, 2} false(0; J; x^{a} e^{- x}; ξ false) false) = trueprefixdet false(double-struckI - ξ K_{N, J} false),

where

double-struckI

is the identity operator and

K_{N, J}

is the integral operator with the kernel acting on

L^{2} false(J false)

. As a consequence, the large N expansion of for

J = (0, s / 4 N)

—referred to as a hard edge scaling—can be deduced from knowledge of the corresponding large N asymptotics of .…”

Section: Functional Form Of Optimal Correction Term For Complex Wishamentioning

confidence: 99%

Finite‐size corrections at the hard edge for the Laguerre β ensemble

Forrester

Trinh

2019

Stud Appl Math

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“…Recently, the thinning and conditioning models have appeared in many situations. For example, gap and conditional probabilities for the thinned unitary ensembles are derived in [6,14]; the asymptotic behavior of mesoscopic fluctuations in the thinned CUE is studied in [3]; the transition between the Tracy-Widom distribution and the Weibull distribution as the probability ω ↓ 0 is considered in [10]; see also [11] for another interesting transition. A nice application in the study of the Riemann zeros can be found in [9].…”

Section: Painlevé XXXIV Universalitymentioning

confidence: 99%

Tracy–Widom Distributions in Critical Unitary Random Matrix Ensembles and the Coupled Painlevé II System

Dai

2018

Commun. Math. Phys.

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We study Fredholm determinants of the Painlevé II and Painlevé XXXIV kernels. In certain critical unitary random matrix ensembles, these determinants describe special gap probabilities of eigenvalues. We obtain Tracy-Widom formulas for the Fredholm determinants, which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled Painlevé II system in dimension four. Moreover, the large gap asymptotics for these Fredholm determinants are derived, where the constant terms are given explicitly in terms of the Riemann zeta-function.

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“…A celebrated toy example is the behaviour near 0 of the squared singular values of Ginibre matrices, also known as the Laguerre Unitary Ensemble [25]. Other examples include non-intersecting squared Bessel paths [17], and the conditional Circular Unitary Ensemble near the edges [5].…”

Section: Introductionmentioning

confidence: 99%

The generating function for the Bessel point process and a system of coupled Painlevé V equations

Charlier

Doeraene

2019

Random Matrices: Theory Appl.

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We study the joint probability generating function for k occupancy numbers on disjoint intervals in the Bessel point process. This generating function can be expressed as a Fredholm determinant. We obtain an expression for it in terms of a system of coupled Painlevé V equations, which are derived from a Lax pair of a Riemann-Hilbert problem. This generalizes a result of Tracy and Widom [24], which corresponds to the case k = 1. We also provide some examples and applications. In particular, several relevant quantities can be expressed in terms of the generating function, like the gap probability on a union of disjoint bounded intervals, the gap between the two smallest particles, and large n asymptotics for n × n Hankel determinants with a Laguerre weight possessing several jumps discontinuities near the hard edge. 2(x − y), α > −1, (1.1)where J α stands for the Bessel function of the first kind of order α (see [22, formula 10.2.2] for a definition of J α ).

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Thinning and conditioning of the circular unitary ensemble

Cited by 40 publications

References 36 publications

Finite‐size corrections at the hard edge for the Laguerre β ensemble

Finite‐size corrections at the hard edge for the Laguerre β ensemble

Tracy–Widom Distributions in Critical Unitary Random Matrix Ensembles and the Coupled Painlevé II System

The generating function for the Bessel point process and a system of coupled Painlevé V equations

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