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THE CHARACTERISTIC POLYNOMIAL OF A RANDOM UNITARY MATRIX: A PROBABILISTIC APPROACH
, 706
"... Abstract. In this paper, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin Fourier transform of such a random polynomial, first obtained by Keating and Snaith in [7], using a simple recursion formula, and from there we ..."
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Cited by 11 (8 self)
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Abstract. In this paper, we propose a probabilistic approach to the study of the characteristic polynomial of a random unitary matrix. We recover the Mellin Fourier transform of such a random polynomial, first obtained by Keating and Snaith in [7], using a simple recursion formula, and from there we are able to obtain the joint law of its radial and angular parts in the complex plane. In particular, we show that the real and imaginary parts of the logarithm of the characteristic polynomial of a random unitary matrix can be represented in law as the sum of independent random variables. From such representations, the celebrated limit theorem obtained by Keating and Snaith in [7] is now obtained from the classical central limit theorems of Probability Theory, as well as some new estimates for the rate of convergence and law of the iterated logarithm type results. 1.
MODGAUSSIAN CONVERGENCE: NEW LIMIT THEOREMS IN PROBABILITY AND NUMBER THEORY
, 807
"... Abstract. We introduce a new type of convergence in probability theory, which we call “modGaussian convergence”. It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of characteristic polynomials or zeta functions. We study thi ..."
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Cited by 7 (4 self)
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Abstract. We introduce a new type of convergence in probability theory, which we call “modGaussian convergence”. It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of characteristic polynomials or zeta functions. We study this type of convergence in detail in the framework of infinitely divisible distributions, and exhibit some unconditional occurrences in number theory, in particular for families of Lfunctions over function fields in the KatzSarnak framework. A similar phenomenon of “modPoisson convergence ” turns out to also appear in the classical ErdősKác Theorem. 1.
ELECTRONIC COMMUNICATIONS in PROBABILITY Theory of Barnes Beta distributions
"... A new family of probability distributions βM,N, M = 0 · · · N, N ∈ N on the unit interval (0, 1] is defined by the Mellin transform. The Mellin transform of βM,N is characterized in terms of products of ratios of Barnes multiple gamma functions, shown to satisfy a functional equation, and a Shinta ..."
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A new family of probability distributions βM,N, M = 0 · · · N, N ∈ N on the unit interval (0, 1] is defined by the Mellin transform. The Mellin transform of βM,N is characterized in terms of products of ratios of Barnes multiple gamma functions, shown to satisfy a functional equation, and a Shintanitype infinite product factorization. The distribution log βM,N is infinitely divisible. If M < N, − log βM,N is compound Poisson, if M = N, log βM,N is absolutely continuous. The integral moments of βM,N are expressed as Selbergtype products of multiple gamma functions. The asymptotic behavior of the Mellin transform is derived and used to prove an inequality involving multiple gamma functions and establish positivity of a class of alternating power series. For application, the Selberg integral is interpreted probabilistically as a transformation of β1,1 into a product of β −1 2,2 s.