Statistical Ensembles of Complex, Quaternion, and Real Matrices

By J. Ginibre. 1965

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Citations in the corpus

Determinantal Point Processes for Machine Learning (2012, background): “Eigenvalues of random matrices [Ginibre, 1965, Mehta and Gaudin, 1960] Let M be a random matrix obtained by drawing every entry independently from the complex normal distribution.”
Geometry of Quantum States: Index (2007, methodology): “For large dimensionality N the unitarity constraints become (relatively) weaker and the non-Hermitian random matrices A are described by the Ginibre ensemble [336, 651]: their spectra cover uniformly the disc of radius 1/ √ N in the complex plane [1015].”
Determinantal Processes and Independence (2005, background): “Ginibre [10] proved that the eigenvalues of Q form a determinantal process in C with the kernel”
The Palm measure and the Voronoi tessellation for the Ginibre process (2006, unknown): “For instance, Le Caer and Ho [16] describe, by means of Monte Carlo simulations, statistical properties of the Voronoi tessellation associated to the Ginibre process of eigenvalues of random complex Gaussian matrices [8] (see also [26]).”
The circular law for random matrices (2007, background): “Ginibre’s [7], in 1965, studied the real, complex and quaternion matrices with i.”
Zeros of Gaussian Analytic Functions and Determinantal Point Processes (2009, background): “Ginibre (29) introduced three ensembles of matrices with i.”
The Noise in the Circular Law and the Gaussian Free Field (2006, background): “2 Facts about the Ginibre ensemble The non-Hermitian matrix ensemble of independent complex Gaussian entries is named for Ginibre who discovered [8] that the joint density of eigenvalues, z1 through zn, is given by dμn(z1) · · · dμn(zn) 1 Zn ∏”
Gaussian fluctuations for non-Hermitian random matrix ensembles (2005, unknown): “The Annals of Probability 2006, Vol. 34, No. 6, 2118–2143 DOI: 10.1214/009117906000000403 © Institute of Mathematical Statistics, 2006”
Resurrecting Recurrent Neural Networks for Long Sequences (2023, methodology): “Fortunately, we can use a classical result from random matrix theory (Ginibre, 1965).”
Zeros of Random Analytic Functions (2006, background): “The Ginibre ensemble has many similarities to the zero set of g.”
BMN Correlators and Operator Mixing in N=4 Super Yang-Mills Theory (2002, methodology): “malization at one-loop order. 29 B Matrix model results on four-point functions Extremal correlators of chiral primaries of the type TrZJ can be calculated for instance by using the result of Ginibre [33], as used already in [8]. For the four-point function the result reads (with 0 <J 1,J 2,J 3,J”
Circular law theorem for random Markov matrices (2008, background): “he uniform law on the centered disc {z ∈ C;|z| ≤ σ} of radius σ. This statement, with the sole ﬁnite positive variance assumption, appears as the culminating point of a long series of partial results [19], [33], [20], [21], [16], [3], [4], [35], [25], [46], [47] concerning the convergence to the Cir”
The Ginibre Ensemble of Real Random Matrices and its Scaling Limits (2008, methodology): “In 1965 J. Ginibre introduced three ensembles of random matrices whose entries were respectively chosen with Gaussian density from R , C and Hamilton’s quaternions [14].”
The single ring theorem (2009, background): “The most famous case of a positive answer to question 1 is the Ginibre ensemble, see [8], and its asymmetric variant, see [18].”
Random matrices: Universality of local spectral statistics of non-Hermitian matrices (2012, methodology): “, for both odd and even n, in [45], [44], [7], [6], [1], [30], [23] (with the (k;l) = (1;0);(0;1) cases worked out previously in [34], [13], [14], building in turn on the foundational work of Ginibre [26]). The precise formulae for these correlation functions are somewhat complicated and involve Pfa”