Asymptotics for Products of Characteristic Polynomials in Classical β-Ensembles

Desrosiers, Patrick; Liu, Dang-Zheng

doi:10.1007/s00365-013-9206-2

Cited by 20 publications

(47 citation statements)

References 64 publications

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“…The derivation relies on an unproved conjecture relating to the asymptotics of a certain generalized hypergeometric function. Such multivariable special functions, introduced into random matrix theory by Constantine and Muirhead in the case β = 1 in the 60's and 70's (see [31]), are finding their way into a number of recent works relating to asymptotics of eigenvalue distributions [7,36,29,18,8,19]. The result of [22] also assumes knowledge of the corresponding asymptotics of E hard β (0; (0, s); βa/2).…”

Section: Introductionmentioning

confidence: 99%

ASYMPTOTICS OF SPACING DISTRIBUTIONS AT THE HARD EDGE FOR Β-Ensembles

Forrester

2013

Random Matrices: Theory Appl.

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In a previous work [J. Math. Phys. 35 (1994) [2539][2540][2541][2542][2543][2544][2545][2546][2547][2548][2549][2550][2551], generalized hypergeometric functions have been used to a give a rigorous derivation of the large s asymptotic form of the general β > 0 gap probability E hard β (0; (0, s); βa/2), provided both βa/2 ∈ Z ≥0 and 2/β ∈ Z + . It is shown how the details of this method can be extended to remove the requirement that 2/β ∈ Z + . Furthermore, a large deviation formula for the gap probability E β (n; (0, x); ME β,N (λ aβ/2 e −2βNλ )) is deduced by writing it in terms of the characteristic function of a certain linear statistic. By scaling x = s/(4N ) 2 and taking N → ∞, this is shown to reproduce a recent conjectured formula for E hard β (n; (0, s); βa/2), βa/2 ∈ Z ≥0 , and moreover to give a prediction without the latter restriction. This extended formula, which for the constant term involves the Barnes double gamma function, is shown to satisfy an asymptotic functional equation relating the gap probability with parameters (β, n, a), to a gap probability with parameters (4/β, n , a ), where n = β(n + 1)/2 − 1, a = β(a − 2)/2 + 2.

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Section: Introductionmentioning

confidence: 99%

ASYMPTOTICS OF SPACING DISTRIBUTIONS AT THE HARD EDGE FOR Β-Ensembles

Forrester

2013

Random Matrices: Theory Appl.

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“…VI 10]. A few results stated here have been proved in the previous paper [22] and will be used later. We finally derive new results on the multivariate functions of Gaussian type and of Airy type.…”

Section: Jack Polynomials and Hypergeometric Functionsmentioning

confidence: 58%

“…where the function on the RHS is the multivariate Airy function in one set of variables, which previously appeared in [18,22]. Asymptotic series of Ai (α) n,m (s; f ) as s j → ±∞ will be given later in Proposition 2.2.…”

Section: Resultsmentioning

confidence: 98%

Scaling Limits of Correlations of Characteristic Polynomials for the Gaussian -Ensemble with External Source

Desrosiers

Liu

2014

International Mathematics Research Notices

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Abstract. We study the averaged product of characteristic polynomials of large random matrices in the Gaussian β-ensemble perturbed by an external source of finite rank. We prove that at the edge of the spectrum, the limiting correlations involve two families of multivariate functions of Airy and Gaussian types. The precise form of the limiting correlations depends on the strength of the nonzero eigenvalues of the external source. A critical value for the latter is obtained and a phase transition phenomenon similar to that of [2] is established. The derivation of our results relies mainly on previous articles by the authors, which deal with duality formulas [18] and asymptotics for Selberg-type integrals [22].

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“…The analogous result for the potential resolvents Q l and P l is the following recursive relation, where z is the distinguished variable 14) and whereby convention U 0 (∅) = 1.…”

Section: Theorem 21 ([43]mentioning

confidence: 93%

“…However the above is just the m = 2 case of (4.13) with the exception of the terms −κz 14) JHEP02 (2015)173 where in their work we identify x → κ, y → 1/N κ. (κN + 1 − 2κ) .…”

Section: Jhep02(2015)173mentioning

confidence: 99%

Loop equation analysis of the circular β ensembles

Witte

Forrester

2015

J. High Energ. Phys.

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We construct a hierarchy of loop equations for invariant circular ensembles. These are valid for general classes of potentials and for arbitrary inverse temperatures Re β > 0 and number of eigenvalues N . Using matching arguments for the resolvent functions of linear statistics f (ζ) = (ζ + z)/(ζ − z) in a particular asymptotic regime, the global regime, we systematically develop the corresponding large N expansion and apply this solution scheme to the Dyson circular ensemble. Currently we can compute the second resolvent function to ten orders in this expansion and also its general Fourier coefficient or moment m k to an equivalent length. The leading large N , large k, k/N fixed form of the moments can be related to the small wave-number expansion of the structure function in the bulk, scaled Dyson circular ensemble, known from earlier work. From the moment expansion we conjecture some exact partial fraction forms for the low k moments. For all of the forgoing results we have made a comparison with the exactly soluble cases of β = 1, 2, 4, general N and even, positive β, N = 2, 3.

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Asymptotics for Products of Characteristic Polynomials in Classical β-Ensembles

Cited by 20 publications

References 64 publications

ASYMPTOTICS OF SPACING DISTRIBUTIONS AT THE HARD EDGE FOR Β-Ensembles

ASYMPTOTICS OF SPACING DISTRIBUTIONS AT THE HARD EDGE FOR Β-Ensembles

Scaling Limits of Correlations of Characteristic Polynomials for the Gaussian -Ensemble with External Source

Loop equation analysis of the circular β ensembles

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