Optimal soft edge scaling variables for the Gaussian and Laguerre even β ensembles

Forrester, Peter J.; Trinh, Allan K.

doi:10.1016/j.nuclphysb.2018.12.006

Cited by 14 publications

(20 citation statements)

References 31 publications

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“…Specifically, summarizing results presented in Ref. for the Laguerre even β ensemble introduce fixed parameters

(λ_{1}, λ, α)

and the auxiliary function

\begin{matrix} true \\ right L_{p} [t^{λ_{1}} e^{- λ t}] (x) & = & left \frac{p! false(N - p false)!}{N!} \int_{0}^{\infty} d t_{1} \dots \int_{0}^{\infty} d t_{N} \prod_{l = 1}^{N} t_{l}^{λ_{1}} e^{- λ t_{l}} {false| x - t_{l} false|}^{α - 1} \\ left \times \prod_{1 ⩽ j < k ⩽ N} {false| t_{k} - t_{j} false|}^{2 λ} e_{p} (x - t_{1}, \dots, x - t_{N}), \end{matrix}

where

e_{p} false(t_{1}, \dots, t_{N} false)

are the elementary symmetric polynomials. Immediately, it can be seen that

L_{N} false[w_{β} (t) false] false(x false) {true|}_{α = n, λ = β / 2} = W_{2 a / β, β, N} {(〈〉, \prod_{l = 1}^{N}, {(x - x_{l})}^{n})}_{L β E (x^{a} e^{- β x / 2})},

where

\begin{matrix} true \\ right W_{} \end{matrix}

…”

Section: Numerics Beyond β=2 General a And A=1 General βmentioning

confidence: 99%

“…Specifically, summarizing results presented in Ref. 17 for the Laguerre even ensemble introduce fixed parameters ( 1 , , ) and the auxiliary function…”

Section: Numerics Beyond = General and = Generalmentioning

confidence: 99%

“…In light of the weak universality observed at the soft edge, in the sense that optimal scaling gives the same order for the leading correction term in all cases where analysis has been possible (see Ref. and references therein), it is natural to consider the large N form of

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

, and related statistical quantities, for other values of the Dyson index β. Our working in Section , using the theory of multivariable hypergeometric functions based on Jack polynomials to analyze the hard edge asymptotics of

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

for general

β > 0

and

a \in {double-struckZ}^{+}

, shows that the extension of to

\frac{s}{4 false(N + a / β false)}

gives an optimal convergence to the limiting distribution at rate

O (N^{- 2})

.…”

Section: Introductionmentioning

confidence: 99%

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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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“…Specifically, summarizing results presented in Ref. for the Laguerre even β ensemble introduce fixed parameters

(λ_{1}, λ, α)

and the auxiliary function

\begin{matrix} true \\ right L_{p} [t^{λ_{1}} e^{- λ t}] (x) & = & left \frac{p! false(N - p false)!}{N!} \int_{0}^{\infty} d t_{1} \dots \int_{0}^{\infty} d t_{N} \prod_{l = 1}^{N} t_{l}^{λ_{1}} e^{- λ t_{l}} {false| x - t_{l} false|}^{α - 1} \\ left \times \prod_{1 ⩽ j < k ⩽ N} {false| t_{k} - t_{j} false|}^{2 λ} e_{p} (x - t_{1}, \dots, x - t_{N}), \end{matrix}

where

e_{p} false(t_{1}, \dots, t_{N} false)

are the elementary symmetric polynomials. Immediately, it can be seen that

L_{N} false[w_{β} (t) false] false(x false) {true|}_{α = n, λ = β / 2} = W_{2 a / β, β, N} {(〈〉, \prod_{l = 1}^{N}, {(x - x_{l})}^{n})}_{L β E (x^{a} e^{- β x / 2})},

where

\begin{matrix} true \\ right W_{} \end{matrix}

…”

Section: Numerics Beyond β=2 General a And A=1 General βmentioning

confidence: 99%

“…Specifically, summarizing results presented in Ref. 17 for the Laguerre even ensemble introduce fixed parameters ( 1 , , ) and the auxiliary function…”

Section: Numerics Beyond = General and = Generalmentioning

confidence: 99%

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

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

for general

β > 0

and

a \in {double-struckZ}^{+}

, shows that the extension of to

\frac{s}{4 false(N + a / β false)}

gives an optimal convergence to the limiting distribution at rate

O (N^{- 2})

.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Forrester

Trinh

2019

Stud Appl Math

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“…We would like to conclude by emphasizing that the recursive approach proposed in this work for the Wishart-Laguerre largest eigenvalue can be obtained from the more general case of the Jacobi ensemble [42,43]. The details of this, and its consequences, are planned for a future work.…”

Section: (17)mentioning

confidence: 93%

Recursion scheme for the largest $\beta$ -Wishart–Laguerre eigenvalue and Landauer conductance in quantum transport

Forrester¹,

Kumar²

2019

J. Phys. A: Math. Theor.

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The largest eigenvalue distribution of the Wishart-Laguerre ensemble, indexed by Dyson parameter β and Laguerre parameter a, is fundamental in multivariate statistics and finds applications in diverse areas. Based on a generalization of the Selberg integral, we provide an effective recursion scheme to compute this distribution explicitly in both the original model, and a fixed-trace variant, for a, β non-negative integers and finite matrix size. For β = 2 this circumvents known symbolic evaluation based on determinants which become impractical for large dimensions. Our exact results have immediate applications in the areas of multiple channel communication and bipartite entanglement. Moreover, we are also led to the exact solution of a long standing problem of finding a general result for Landauer conductance distribution in a chaotic mesoscopic cavity with two ideal leads. Thus far, exact closed-form results for this were available only in the Fourier-Laplace space or could be obtained on a case-by-case basis.

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“…In special cases, these scalings are quantified in [39] for the hard edge limit, and in [30] for the soft edge limit. In the recent works [24,25], recursions applying for the β-Laguerre ensemble have been used to exhibit the optimal rates beyond the classical cases β = 1, 2 and 4.…”

Section: Direct Computation Of P (N)mentioning

confidence: 99%

Computable structural formulas for the distribution of the $β$-Jacobi edge eigenvalues

Forrester¹,

Kumar²

2020

Preprint

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The Jacobi ensemble is one of the classical ensembles of random matrix theory. Prominent in applications are properties of the eigenvalues at the spectrum edge, specifically the distribution of the largest (e.g. Roy's largest root test in multivariate statistics) and smallest (e.g. condition numbers of linear systems) eigenvalues. We identify three ranges of parameter values for which the gap probability determining these distributions is a finite sum with respect to particular bases, and moreover make use of a certain differential-difference system fundamental in the theory of the Selberg integral to provide a recursive scheme to compute the corresponding coefficients.

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Optimal soft edge scaling variables for the Gaussian and Laguerre even β ensembles

Cited by 14 publications

References 31 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

Recursion scheme for the largest $\beta$ -Wishart–Laguerre eigenvalue and Landauer conductance in quantum transport

Computable structural formulas for the distribution of the $β$-Jacobi edge eigenvalues

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