Products of random matrices from fixed trace and induced Ginibre ensembles

Akemann, Gernot; Cikovic, Milan

doi:10.1088/1751-8121/aab8a9

Cited by 7 publications

(6 citation statements)

References 85 publications

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“…The same holds true for the kernel of skew orthogonal polynomials in Pfaffian process including the real Ginibre ensemble [5]. Combined with the result for the kernel K (M ) N of products of M independent complex Ginibre matrices [4], extended to non homogeneous variances in [2], we have the following statement…”

Section: Introduction and Main Resultssupporting

confidence: 60%

Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality

Akemann

Götze

Neuschel

2021

Electron. Commun. Probab.

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We compute the average characteristic polynomial of the Hermitised product of M real or complex non-Hermitian Wigner matrices of size N × N with i.i.d. matrix elements, and the average of the characteristic polynomial of a product of M such matrices times the characteristic polynomial of the conjugate product matrix. Surprisingly, the results agree with that of the product of M real or complex Ginibre matrices at finite-N , which have i.i.d. Gaussian entries. For the latter the average characteristic polynomial yields the orthogonal polynomial for the singular values of the product matrix, whereas the product of the two characteristic polynomials involves the kernel of complex eigenvalues. This extends the result of Forrester and Gamburd for one characteristic polynomial of such a single random matrix and only depends on the first two moments. In the limit M → ∞ at fixed N we determine the locations of the zeros of a single characteristic polynomial, rescaled as Lyapunov exponents by taking the logarithm of the M th root. The position of the jth zero agrees asymptotically for large-j with the position of the jth Lyapunov exponent for products of Gaussian random matrices, hinting at the universality of the latter.

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Section: Introduction and Main Resultssupporting

confidence: 60%

Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality

Akemann

Götze

Neuschel

2021

Electron. Commun. Probab.

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“…We assume in Algorithm 4.4 that this condition holds, and we refer to [26, Section VII.1.4] and [35] for more details. (2) x m+ 1 2 = x m + ∆ty m+ 1 4 ; (3) compute the Lagrange multiplier θ m ∈ R m associated with x m+ 1 2 to enforce the constraint, using Algorithm 4.6 below (if convergence has been reached);…”

Section: Constrained Langevin Dynamicsmentioning

confidence: 99%

Coulomb Gases Under Constraint: Some Theoretical and Numerical Results

Chafäı¹,

Ferré²,

Stoltz³

2021

SIAM J. Math. Anal.

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We consider Coulomb gas models for which the empirical measure typically concentrates, when the number of particles becomes large, on an equilibrium measure minimizing an electrostatic energy. We study the behavior when the gas is conditioned on a rare event. We first show that the special case of quadratic confinement and linear constraint is exactly solvable due to a remarkable factorization, and that the conditioning has then the simple effect of shifting the cloud of particles without deformation. To address more general cases, we perform a theoretical asymptotic analysis relying on a large deviations technique known as the Gibbs conditioning principle. The technical part amounts to establishing that the conditioning ensemble is an Icontinuity set of the energy. This leads to characterizing the conditioned equilibrium measure as the solution of a modified variational problem. For simplicity, we focus on linear statistics and on quadratic statistics constraints. Finally, we numerically illustrate our predictions and explore cases in which no explicit solution is known. For this, we use a Generalized Hybrid Monte Carlo algorithm for sampling from the conditioned distribution for a finite but large system. Contents 12 4.1. Description of the algorithm 12 4.2. Numerical results 17 Acknowledgements 20 Appendix A.

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“…Pointwise convergence of the density ρ m n ofμ n orμ m n has been also discussed in [2,4,35]. In particular Akemann and Burda describe the asymptotic…”

Section: The Optimal Rate Of Convergence For Products Of Ginibre Matricesmentioning

confidence: 99%

Rate of convergence for products of independent non-Hermitian random matrices

Jalowy

2021

Electron. J. Probab.

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We study the rate of convergence of the empirical spectral distribution of products of independent non-Hermitian random matrices to the power of the Circular Law. The distance to the deterministic limit distribution will be measured in terms of a uniform Kolmogorov-like distance. First, we prove that for products of Ginibre matrices, the optimal rate is given by O(1/ √ n), which is attained with overwhelming probability up to a logarithmic correction. Avoiding the edge, the rate of convergence of the mean empirical spectral distribution is even faster. Second, we show that also products of matrices with independent entries attain this optimal rate in the bulk up to a logarithmic factor. In the case of Ginibre matrices, we apply a saddlepoint approximation to a double contour integral representation of the density and in the case of matrices with independent entries we make use of techniques from local laws.

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Products of random matrices from fixed trace and induced Ginibre ensembles

Cited by 7 publications

References 85 publications

Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality

Characteristic polynomials of products of non-Hermitian Wigner matrices: finite-N results and Lyapunov universality

Coulomb Gases Under Constraint: Some Theoretical and Numerical Results

Rate of convergence for products of independent non-Hermitian random matrices

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