Selberg Integral as a Meromorphic Function

Ostrovsky, Dmitry

doi:10.1093/imrn/rns170

Cited by 26 publications

(97 citation statements)

References 33 publications

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“…Thanks to the computations performed by Ostrovsky [23], we can also state our main result in the following equivalent way:…”

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confidence: 83%

The distribution of Gaussian multiplicative chaos on the unit interval

Rémy¹,

Zhu²

2020

Ann. Probab.

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We consider a sub-critical Gaussian multiplicative chaos (GMC) measure defined on the unit interval [0, 1] and prove an exact formula for the fractional moments of the total mass of this measure. Our formula includes the case where log-singularities (also called insertion points) are added in 0 and 1, the most general case predicted by the Selberg integral. The idea to perform this computation is to introduce certain auxiliary functions resembling holomorphic observables of conformal field theory that will be solutions of hypergeometric equations. Solving these equations then provides non-trivial relations that completely determine the moments we wish to compute. We also include a detailed discussion of the so-called reflection coefficients appearing in tail expansions of GMC measures and in Liouville theory. Our theorem provides an exact value for one of these coefficients. Lastly we mention some additional applications to small deviations for GMC measures, to the behavior of the maximum of the log-correlated field on the interval and to random hermitian matrices. †, ‡ Research supported in part by the ANR grant Liouville (ANR-15-CE40-0013). MSC 2010 subject classifications: Primary 60G57; secondary 60G15, 60G60, 60G70, 82B23.1 1 Our normalization differs from the ln 1 |x−y| usually found in the literature.

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“…Thanks to the computations performed by Ostrovsky [23], we can also state our main result in the following equivalent way:…”

mentioning

confidence: 83%

The distribution of Gaussian multiplicative chaos on the unit interval

Rémy¹,

Zhu²

2020

Ann. Probab.

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“…However, it does not seem that such simple Beta factorizations always exist for n ≥ 3. See (6) in [57] for a related identity, and also [44] for another point of view on (25), where s is interpreted as a parameter of a so-called Barnes Beta distribution.…”

Section: 4mentioning

confidence: 99%

Some properties of the free stable distributions

Simon¹,

Wang²

2020

Ann. Inst. H. Poincaré Probab. Statist.

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We investigate certain analytical properties of the free α−stable densities on the line. We prove that they are all classically infinitely divisible when α ≤ 1, and that they belong to the extended Thorin class when α ≤ 3/4. The Lévy measure is explicitly computed for α = 1, showing that the free 1-stable random variables are not Thorin except in the drifted Cauchy case. In the symmetric case we show that the free stable densities are not infinitely divisible when α > 1. In the one-sided case we prove, refining unimodality, that the densities are whale-shaped that is their successive derivatives vanish exactly once. Finally, we derive a collection of results connected to the fine structure of the one-sided free stable densities, including a detailed analysis of the Kanter random variable, complete asymptotic expansions at zero, a new identity for the Beta-Gamma algebra, and several intrinsic properties of whale-shaped densities.where X 1 , X 2 are free independent copies of a random variable X with density f , a, b are arbitrary positive real numbers, c is a positive real number depending on a, b, and d is a real number. As in the classical framework, it turns out that there exist solutions to (1) only if c = (a α + b α ) 1/α for some fixed α ∈ (0, 2]. We will be mostly concerned with free strictly stable densities, which correspond to the case d = 0. In this framework the Voiculescu transform of f writes, up to multiplicative normalization,We refer e.g. to [43] for some background on the free additive convolution, to [11] for the original solution to the equation (1), and to the introduction of [29] for the above parametrization (α, ρ), which mimics that of the strict classical framework. Let us also recall that free stable laws appear as limit distributions of spectra of large random matrices with possibly unbounded variance -see [10,18], and that their domains of attraction have been fully characterized in [12,13]. In the following, we will denote by X α,ρ the random variable whose Voiculescu transform is given by (2), and set f α,ρ for its density. The analogy with the classical case extends to the fact, observed in Corollary 1.3 of [29], that with our parametrization one hasThe explicit form of the Voiculescu transform also shows that X α,ρ d = −X α,1−ρ . In this paper, some focus will put on the one-sided case and we will use the shorter notations X α,1 = X α and f α,1 = f α .Throughout, the random variable X α,ρ will be mostly handled as a classical random variable via its 2010 Mathematics Subject Classification. 60E07; 46L54; 62E15; 33E20.

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“…(4.11) In the case that a ∈ Z ≥0 , simple manipulation of the product and use of Stirling's formula shows that the large N form of (4.11) is To compute the large N form of (4.11) without the assumption a ∈ Z ≥0 , we follow the lead of the recent works [25,4,32] and introduce the Barnes double gamma function Γ 2 (z; 1, τ). This function is related to the usual gamma function through the two functional equations…”

Section: Corollary 1 Up To Terms O(1/n ) We Havementioning

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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Selberg Integral as a Meromorphic Function

Cited by 26 publications

References 33 publications

The distribution of Gaussian multiplicative chaos on the unit interval

The distribution of Gaussian multiplicative chaos on the unit interval

Some properties of the free stable distributions

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

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