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On the Eigenvalues of Random Matrices
Journal of Applied Probability, 1994Let M be a random matrix chosen from Haar measure on the unitary group Un. Let Z = X + iY be a standard complex normal random variable with X and Y independent, mean 0 and variance ½ normal variables. We show that for j = 1, 2, …, Tr(Mj) are independent and distributed as √jZ asymptotically as n →∞.
Diaconis, Persi, Shahshahani, Mehrdad
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Stochastic Models, 2006
A class of infinitely divisible covariance mixtures of Gaussian random matrices is introduced, and a characterization within the class of infinitely divisible left-orthogonally invariant matrix distributions is proved.
Barndorff-Nielsen, Ole Eiler +2 more
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A class of infinitely divisible covariance mixtures of Gaussian random matrices is introduced, and a characterization within the class of infinitely divisible left-orthogonally invariant matrix distributions is proved.
Barndorff-Nielsen, Ole Eiler +2 more
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On the rank of random matrices
Random Structures and Algorithms, 2000The main result of the paper concerns \(n\times n\) random matrices \(M\) with independent identically distributed entries \(m_{ij}\) such that \(\text{Pr} \{m_{ij}=1\} =p(n)\) and \(\text{Pr}\{m_{ij} =0\}=1-p(n)\). To answer the question posed by \textit{J. Blömer}, \textit{R. Karp}, and \textit{E. Welzl} [Random Struct. Algorithms 10, No. 4, 407-419 (
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Kibernetyka ta Systemnyi Analiz
The paper examines methods for assessing the distribution of elements in a stochastic matrix assuming an exponential distribution of elements in the corresponding adjacency matrix of a graph. Two cases are considered: the first assumes homogeneity of all graph vertices, while the second assumes heterogeneity in the distribution of vertices with ...
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The paper examines methods for assessing the distribution of elements in a stochastic matrix assuming an exponential distribution of elements in the corresponding adjacency matrix of a graph. Two cases are considered: the first assumes homogeneity of all graph vertices, while the second assumes heterogeneity in the distribution of vertices with ...
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Random correlation matrices [PDF]
Australas. J Comb., 1998 The paper concerns the number \(\Lambda\) of nonzero elements of a doubly stochastic matrix \(D\) whose entries \(P(i,j)\) are squares of the correlation coefficient \(C(\cdot,\cdot)\) between a nonzero linear combination, specified by \(j\), of the component Boolean functions of \(F= (f_1,\dots, f_n)\) and a linear Boolean function of \(X= (x_1,\dots,
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1963
Let \(P(n,N(n))\) denote the probability that a random \(n\) by \(n\) matrix with \(N(n)\) 1's and \(n^2-N(n)\) 0's has a positive permanent. The authors show that if \(N(n)=n\log n+cn+o(n)\), where \(c\) is an arbitrary constant, then \(\lim_{n \to \infty} P(n,N(n)) = \exp(-2e^{-c})\).
Erdős, Pál, Rényi, Alfréd
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Let \(P(n,N(n))\) denote the probability that a random \(n\) by \(n\) matrix with \(N(n)\) 1's and \(n^2-N(n)\) 0's has a positive permanent. The authors show that if \(N(n)=n\log n+cn+o(n)\), where \(c\) is an arbitrary constant, then \(\lim_{n \to \infty} P(n,N(n)) = \exp(-2e^{-c})\).
Erdős, Pál, Rényi, Alfréd
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Oberwolfach Reports, 2020
Large complex systems tend to develop universal patterns that often represent their essential characteristics. For example, the cumulative effects of independent or weakly dependent random variables often yield the Gaussian universality class via the central limit theorem. For non-commutative random variables, e.g.
László Erdős +2 more
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Large complex systems tend to develop universal patterns that often represent their essential characteristics. For example, the cumulative effects of independent or weakly dependent random variables often yield the Gaussian universality class via the central limit theorem. For non-commutative random variables, e.g.
László Erdős +2 more
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SIAM Review, 1967
Introduction. It has been observed repeatedly that von iNeumann made important contributions to almost all parts of mathematics with the exception of number theory. He had a particular interest in those parts of mathematics which formed cornerstones of other, more empirical sciences, such as physics or economics.
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Introduction. It has been observed repeatedly that von iNeumann made important contributions to almost all parts of mathematics with the exception of number theory. He had a particular interest in those parts of mathematics which formed cornerstones of other, more empirical sciences, such as physics or economics.
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Random Matrices and Brownian Motion
Combinatorics, Probability and Computing, 1993For T ∈ GLn (Fq), let Ωn (t, T) be the number of irreducible factors of degree less than or equal to nt in the characteristic polynomial of T. Letand suppose T is chosen from G Ln(Fq) at random uniformly. We prove that the stochastic process ≺Zn(t)≻t∈[0, 1] converges to the standard Brownian motion process W(t), as n → ∞.
William M. Y. Goh, Eric Schmutz
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