Symmetries of the quaternionic Ginibre ensemble

Dubach, Guillaume

doi:10.1142/s2010326321500131

Cited by 15 publications

(25 citation statements)

References 16 publications

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“…as noted already in [17]. As in the complex case [41], due to the bi-orthogonality (2.6) that holds for indices with or without conjugation, we have the following representations of the identity matrix I N as a dyadic product: 12) which is also called closure relation.…”

Section: )mentioning

confidence: 90%

“…1.2.4]. Note also that the overlap matrix O ij is Hermitian, as mentioned in [17]. Because the overlaps O ij depend on all N complex conjugated pairs we introduce the following notation.…”

Section: )mentioning

confidence: 99%

“…This leads us to the conclusion that self-overlaps for conjugate eigenvalues O ii and O ii , are identical for each i, cf. [17], as well as O ij and O ij for each pair i, j. Second, due to the permutation symmetry we may focus on the index pairs i, j ∈ {1, 1, 2, 2} only.…”

Section: Quaternionic Random Matrix Ensembles and Mean Overlapsmentioning

confidence: 99%

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Universal eigenvector correlations in quaternionic Ginibre ensembles

Akemann

Förster

Kieburg

2020

J. Phys. A: Math. Theor.

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Non-Hermitian random matrices enjoy non-trivial correlations in the statistics of their eigenvectors. We study the overlap among left and right eigenvectors in Ginibre ensembles with quaternion valued Gaussian matrix elements. This concept was introduced by Chalker and Mehlig in the complex Ginibre ensemble. Using a Schur decomposition, for harmonic potentials we can express the overlap in terms of complex eigenvalues only, coming in conjugate pairs in this symmetry class. Its expectation value leads to a Pfaffian determinant, for which we explicitly compute the matrix elements for the induced Ginibre ensemble with 2α zero eigenvalues, for finite matrix size N . In the macroscopic large-N limit in the bulk of the spectrum we recover the limiting expressions of the complex Ginibre ensemble for the diagonal and off-diagonal overlap, which are thus universal.

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Section: )mentioning

confidence: 90%

“…1.2.4]. Note also that the overlap matrix O ij is Hermitian, as mentioned in [17]. Because the overlaps O ij depend on all N complex conjugated pairs we introduce the following notation.…”

Section: )mentioning

confidence: 99%

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Universal eigenvector correlations in quaternionic Ginibre ensembles

Akemann

Förster

Kieburg

2020

J. Phys. A: Math. Theor.

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“…a similar expression holds, see (2.16) below. The same mechanism applies to the overlaps in the quaternionic Ginibre ensemble [20,21]. In the real ensemble [23] the Laplace transformed joint density of overlap and conditional eigenvalue is given by an averaged ratio of characteristic polynomials, thus only depending on the eigenvalues too.…”

Section: )mentioning

confidence: 91%

“…These, as well as truncated unitary and spherical ensembles, were analysed in [16] using probabilistic means, after an earlier breakthrough for these methods in [17], see also [18] for the correlations between angles of eigenvectors. The quaternionic Ginibre ensemble appeared more recently from a probabilistic angle [19,20] as well as for finite-N in [21], using the heuristic tools of [22]. An entirely different approach uses supersymmetry [23] or orthogonal polynomials [24], expressing the relevant quantities in terms of expectation values of characteristic polynomials.…”

Section: Introductionmentioning

confidence: 99%

On the determinantal structure of conditional overlaps for the complex Ginibre ensemble

Akemann

Tribe

Tsareas

et al. 2019

Random Matrices: Theory Appl.

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In these proceedings we summarise how the determinantal structure for the conditional overlaps among left and right eigenvectors emerges in the complex Ginibre ensemble at finite matrix size. An emphasis is put on the underlying structure of orthogonal polynomials in the complex plane and its analogy to the determinantal structure of k-point complex eigenvalue correlation functions. The off-diagonal overlap is shown to follow from the diagonal overlap conditioned on k ≥ 2 complex eigenvalues. As a new result we present the local bulk scaling limit of the conditional overlaps away from the origin. It is shown to agree with the limit at the origin and is thus universal within this ensemble.

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Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

Fyodorov

Tarnowski

2020

Ann. Henri Poincaré

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We study the distribution of the eigenvalue condition numbers $$\kappa _i=\sqrt{ ({\mathbf{l}}_i^* {\mathbf{l}}_i)({\mathbf{r}}_i^* {\mathbf{r}}_i)}$$ κ i = ( l i ∗ l i ) ( r i ∗ r i ) associated with real eigenvalues $$\lambda _i$$ λ i of partially asymmetric $$N\times N$$ N × N random matrices from the real Elliptic Gaussian ensemble. The large values of $$\kappa _i$$ κ i signal the non-orthogonality of the (bi-orthogonal) set of left $${\mathbf{l}}_i$$ l i and right $${\mathbf{r}}_i$$ r i eigenvectors and enhanced sensitivity of the associated eigenvalues against perturbations of the matrix entries. We derive the general finite N expression for the joint density function (JDF) $${{\mathcal {P}}}_N(z,t)$$ P N ( z , t ) of $$t=\kappa _i^2-1$$ t = κ i 2 - 1 and $$\lambda _i$$ λ i taking value z, and investigate its several scaling regimes in the limit $$N\rightarrow \infty $$ N → ∞ . When the degree of asymmetry is fixed as $$N\rightarrow \infty $$ N → ∞ , the number of real eigenvalues is $$\mathcal {O}(\sqrt{N})$$ O ( N ) , and in the bulk of the real spectrum $$t_i=\mathcal {O}(N)$$ t i = O ( N ) , while on approaching the spectral edges the non-orthogonality is weaker: $$t_i=\mathcal {O}(\sqrt{N})$$ t i = O ( N ) . In both cases the corresponding JDFs, after appropriate rescaling, coincide with those found in the earlier studied case of fully asymmetric (Ginibre) matrices. A different regime of weak asymmetry arises when a finite fraction of N eigenvalues remain real as $$N\rightarrow \infty $$ N → ∞ . In such a regime eigenvectors are weakly non-orthogonal, $$t=\mathcal {O}(1)$$ t = O ( 1 ) , and we derive the associated JDF, finding that the characteristic tail $${{\mathcal {P}}}(z,t)\sim t^{-2}$$ P ( z , t ) ∼ t - 2 survives for arbitrary weak asymmetry. As such, it is the most robust feature of the condition number density for real eigenvalues of asymmetric matrices.

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Symmetries of the quaternionic Ginibre ensemble

Cited by 15 publications

References 16 publications

Universal eigenvector correlations in quaternionic Ginibre ensembles

Universal eigenvector correlations in quaternionic Ginibre ensembles

On the determinantal structure of conditional overlaps for the complex Ginibre ensemble

Condition Numbers for Real Eigenvalues in the Real Elliptic Gaussian Ensemble

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