On the Barnes function

Adamchik, Victor

doi:10.1145/384101.384104

Cited by 33 publications

(52 citation statements)

References 11 publications

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“…The Barnes G-function, which was first introduced by Barnes in [3] (see also [1]), may be defined via its infinite product representation where γ is the Euler constant. From (1.1), one can easily deduce the following (useful) development of the logarithm of G (1 + z) for |z| < 1:…”

Section: Introduction Motivation and Main Resultsmentioning

confidence: 99%

The Barnes $G$ function and its relations with sums and products of generalized Gamma convolution variables

Nikeghbali

Yor²

2009

Electron. Commun. Probab.

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The barnes G function and its relations with sums and products of generalized gamma convolution variables The barnes G function and its relations with sums and products of generalized gamma convolution variables AbstractWe give a probabilistic interpretation for the Barnes G-function which appears in random matrix theory and in analytic number theory in the important moments conjecture due to Keating-Snaith for the Riemann zeta function, via the analogy with the characteristic polynomial of random unitary matrices. We show that the Mellin transform of the characteristic polynomial of random unitary matrices and the Barnes G-function are intimately related with products and sums of gamma, beta and log-gamma variables. In particular, we show that the law of the modulus of the characteristic polynomial of random unitary matrices can be expressed with the help of products of gamma or beta variables. This leads us to prove some non standard type of limit theorems for the logarithmic mean of the so called generalized gamma convolutions. AbstractWe give a probabilistic interpretation for the Barnes G-function which appears in random matrix theory and in analytic number theory in the important moments conjecture due to Keating-Snaith for the Riemann zeta function, via the analogy with the characteristic polynomial of random unitary matrices. We show that the Mellin transform of the characteristic polynomial of random unitary matrices and the Barnes G-function are intimately related with products and sums of gamma, beta and log-gamma variables. In particular, we show that the law of the modulus of the characteristic polynomial of random unitary matrices can be expressed with the help of products of gamma or beta variables. This leads us to prove some non standard type of limit theorems for the logarithmic mean of the so called generalized gamma convolutions.

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Section: Introduction Motivation and Main Resultsmentioning

confidence: 99%

The Barnes $G$ function and its relations with sums and products of generalized Gamma convolution variables

Nikeghbali

Yor²

2009

Electron. Commun. Probab.

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“…Incidentally, the numbers χ k,0 = B 2k /(2k(2k − 2)) which appear in (92) are precisely the virtual Euler characteristics for the moduli space of unpunctured Riemann surfaces [31]. To relate the values of the Barnes G function on the negative axis to the values on the positive axis we use the following expression [32],…”

Section: The Large N Limit Of the Free Energymentioning

confidence: 99%

Fine structure in the large n limit of the non-Hermitian Penner matrix model

2015

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In this paper we apply results on the asymptotic zero distribution of the Laguerre polynomials to discuss generalizations of the standard large n limit in the non-hermitian Penner matrix model. In these generalizations g n n → t, but the product g n n is not necessarily fixed to the value of the 't Hooft coupling t. If t > 1 and the limit l = lim n→∞ | sin(π/g n )| 1/n exists, then the large n limit is well-defined but depends both on t and on l. This result implies that for t > 1 the standard large n limit with g n n = t fixed is not well-defined. The parameter l determines a fine structure of the asymptotic eigenvalue support: for l = 0 the support consists of an interval on the real axis with charge fraction Q = 1 − 1/t and an l-dependent oval around the origin with charge fraction 1/t. For l = 1 these two components meet, and for l = 0 the oval collapses to the origin. We also calculate the total electrostatic energy E, which turns out to be independent of l, and the free energy F = E − Q ln l, which does depend of the fine structure parameter l. The existence of large n asymptotic expansions of F beyond the planar limit as well as the double-scaling limit are also discussed.

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“…We can expand these expressions in powers of 1/L by rewriting the product of factorials in terms of Barnes G-functions [1,2,[4][5][6] and using the asymptotic expansion of the Barnes G-function, see e.g. Ref.…”

Section: Generating High Precision Numerical Datamentioning

confidence: 99%

“…From that conjecture they were able to relate the generating function for the probability distribution to the determinant (1) [19]. Defining φ(L, θ) ≡ L/2 m=0 P (L, m)2 m cos m (θ) (2) and…”

Section: Introductionmentioning

confidence: 99%

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Exact asymptotics of the characteristic polynomial of the symmetric Pascal matrix

Mitra

2009

Journal of Combinatorial Theory, Series A

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We have obtained the exact asymptotics of the determinant det 1 r,s L r+s−2 r−1 + exp(iθ)δ r,s . Inverse symbolic computing methods were used to obtain exact analytical expressions for all terms up to relative order L −14 to the leading term. This determinant is known to give weighted enumerations of cyclically symmetric plane partitions, weighted enumerations of certain families of vicious walkers and it has been conjectured to be proportional to the one point function of the O(1) loop model on a cylinder of circumference L. We apply our result to the loop model and give exact expressions for the asymptotics of the average of the number of loops surrounding a point and the fluctuation in this number. For the related bond percolation model at the critical point, we give exact expressions for the asymptotics of the probability that a point is on a cluster that wraps around a cylinder of even circumference and the probability that a point is on a cluster spanning a cylinder of odd circumference.

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On the Barnes function

Cited by 33 publications

References 11 publications

The Barnes $G$ function and its relations with sums and products of generalized Gamma convolution variables

The Barnes $G$ function and its relations with sums and products of generalized Gamma convolution variables

Fine structure in the large n limit of the non-Hermitian Penner matrix model

Exact asymptotics of the characteristic polynomial of the symmetric Pascal matrix

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