Density of Small Singular Values of the Shifted Real Ginibre Ensemble

@article{Cipolloni2022DensityOS,
  title={Density of Small Singular Values of the Shifted Real Ginibre Ensemble},
  author={Giorgio Cipolloni and L{\'a}szl{\'o} Erdős and Dominik Schr{\"o}der},
  journal={Annales Henri Poincar{\'e}},
  year={2022}
}
We derive a precise asymptotic formula for the density of the small singular values of the real Ginibre matrix ensemble shifted by a complex parameter z as the dimension tends to infinity. For z away from the real axis the formula coincides with that for the complex Ginibre ensemble we derived earlier in Cipolloni et al. (Prob Math Phys 1:101–146, 2020). On the level of the one-point function of the low lying singular values we thus confirm the transition from real to complex Ginibre ensembles… 
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References

SHOWING 1-10 OF 25 REFERENCES
On the condition number of the shifted real Ginibre ensemble
TLDR
An accurate lower tail estimate is derived on the lowest singular value σ1(X − z) of a real Gaussian (Ginibre) random matrixX shifted by a complex parameter z and an improved upper bound on the eigenvalue condition numbers for real Ginibre matrices is obtained.
Universal Signature from Integrability to Chaos in Dissipative Open Quantum Systems.
We study the transition between integrable and chaotic behavior in dissipative open quantum systems, exemplified by a boundary driven quantum spin chain. The repulsion between the complex eigenvalues
The Ginibre Ensemble of Real Random Matrices and its Scaling Limits
We give a closed form for the correlation functions of ensembles of a class of asymmetric real matrices in terms of the Pfaffian of an antisymmetric matrix formed from a 2 × 2 matrix kernel
Statistics of real eigenvalues in Ginibre's ensemble of random real matrices.
The integrable structure of Ginibre's orthogonal ensemble of random matrices is looked at through the prism of the probability p(n,k) to find exactly k real eigenvalues in the spectrum of an n x n
On Statistics of Bi-Orthogonal Eigenvectors in Real and Complex Ginibre Ensembles: Combining Partial Schur Decomposition with Supersymmetry
  • Y. Fyodorov
  • Mathematics
    Communications in Mathematical Physics
  • 2018
We suggest a method of studying the joint probability density (JPD) of an eigenvalue and the associated ‘non-orthogonality overlap factor’ (also known as the ‘eigenvalue condition number’) of the
Fluctuation around the circular law for random matrices with real entries
We extend our recent result [Cipolloni, Erdős, Schroder 2019] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices $X$ with independent, identically distributed
Eigenvalue statistics of the real Ginibre ensemble.
TLDR
A computationally tractable formula for the cumulative probability density of the largest real eigenvalue is presented, relevant to May's stability analysis of biological webs.
Eigenvalue statistics of random real matrices.
  • Lehmann, Sommers
  • Computer Science, Mathematics
    Physical review letters
  • 1991
TLDR
The joint probability density of eigenvalues in a Gaussian ensemble of real asymmetric matrices, which is invariant under orthogonal transformations is determined, which indicates thatrices of the type considered appear in models for neural-network dynamics and dissipative quantum dynamics.
Superbosonization of Invariant Random Matrix Ensembles
Abstract‘Superbosonization’ is a new variant of the method of commuting and anti-commuting variables as used in studying random matrix models of disordered and chaotic quantum systems. We here give a
The circular law
The circular law theorem states that the empirical spectral distribution of a n×n random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as
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