Symmetric Function Theory and Unitary Invariant Ensembles

Jonnadula, Bhargavi; Keating, Jonathan P.; Mezzadri, Francesco

doi:10.48550/arxiv.2003.02620

Cited by 6 publications

(10 citation statements)

References 30 publications

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“…For Hermitian ensembles, Jonnadula, Keating, and Mezzadri [99,100] proved a version of proposition 3.9 using multivariate orthogonal polynomials. They subsequently calculated the asymptotics of the moments of the associated characteristic polynomials, and , in the GUE case, identified a subtle dependence on the parity of the matrix size.…”

Section: Definition 319 (Symplectic Patterns)mentioning

confidence: 99%

Maxima of log-correlated fields: some recent developments

Bailey,

Keating

2021

Preprint

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We dedicate this review to the memory of Fritz Haake, who did so much to understand and explain the connections between Random Matrix Theory and Quantum Chaos, and whose friendship one of us (JPK) was greatly fortunate to enjoy.ABSTRACT. We review recent progress relating to the extreme value statistics of the characteristic polynomials of random matrices associated with the classical compact groups, and of the Riemann zeta-function and other L-functions, in the context of the general theory of logarithmicallycorrelated Gaussian fields. In particular, we focus on developments related to the conjectures of Fyodorov & Keating concerning the extreme value statistics, moments of moments, connections to Gaussian Multiplicative Chaos, and explicit formulae derived from the theory of symmetric functions. CONTENTSBy convention, ∆(z 1 ) = 1.

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Section: Definition 319 (Symplectic Patterns)mentioning

confidence: 99%

Maxima of log-correlated fields: some recent developments

Bailey,

Keating

2021

Preprint

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“…µ form a basis for symmetric polynomials of degree |µ|. The Schur polynomials can be expanded as [35]…”

Section: Young Diagram Of µmentioning

confidence: 99%

On the moments of characteristic polynomials

Jonnadula¹,

Keating²,

Mezzadri³

2021

Preprint

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We examine the asymptotics of the moments of characteristic polynomials of N ×N matrices drawn from the Hermitian ensembles of Random Matrix Theory, in the limit as N → ∞. We focus in particular on the Gaussian Unitary Ensemble, but discuss other Hermitian ensembles as well. We employ a novel approach to calculate asymptotic formulae for the moments, enabling us to uncover subtle structure not apparent in previous approaches.

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“…As noticed in the previously cited papers, after the proper rescaling to set N − k variables to 1, one ends up with g λ (x) = S λ x + 1 N −1 . Determinantal manipulations often use the minor expansion used in remark 4.26, itself equivalent to the partial fraction expansion of t → H[tX] or t → H[t(X − Y )] given in (161), but to perform an asymptotic analysis, it is better suited to use an integral form coming from a residue sum ; in our setting, this means going from (162) to (159). Such a manipulation is for instance done in [146, (3), (4)] starting from the residue sum of [228,Prop.…”

Section: Asymptotic Representation Theorymentioning

confidence: 99%

“…On the other hand, formula (22) and the underlying philosophy of proof can be extended in a straightforward way to the orthogonal and symplectic circular ensembles with a zonal polynomial or more generally to circular β ensembles, replacing the Schur function by a Jack polynomial expansion (see § 3.3). In fact, the philosophy could also be applied to any ensemble, if one defines the correct equivalent of rectangular Schur function as in [159], and it could even be generalised to other random variables associated to more general symmetric functions such as the Hall-Littlewood ones (see remark 3.1).…”

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confidence: 99%

A new approach to the characteristic polynomial of a random unitary matrix

Barhoumi-Andréani¹

2020

Preprint

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Since the seminal work of Keating and Snaith, the characteristic polynomial of a random Haar-distributed unitary matrix has seen several of its functional studied or turned into a conjecture; for instance:• its value in 1 (Keating-Snaith theorem),• the truncation of its Fourier series up to any fraction of its degree,• the computation of the relative volume of the Birkhoff polytope,• its products and ratios taken in different points,• the product of its iterated derivatives in different points,• functionals in relation with sums of divisor functions in F q [X].• its mid-secular coefficients,• the "moments of moments", etc.We revisit or compute for the first time the asymptotics of the integer moments of these last functionals and several others. The method we use is a very general one based on reproducing kernels, a symmetric function generalisation of some classical orthogonal polynomials interpreted as the Fourier transform of particular random variables and a local Central Limit Theorem for these random variables. We moreover provide an equivalent paradigm based on a randomisation of the mid-secular coefficients to rederive them all. These methodologies give a new and unified framework for all the considered limits and explain the apparition of Hankel determinants or Wronskians in the limiting functional. Y. BARHOUMI-ANDRÉANI 2. Notations and prerequisites 2.1. General notations and conventions 2.2. Reminders on symmetric functions 2.3. Scalar products and unitary integrals 2.4. Classical tricks 3. Theory 3.1. The ubiquitous Schur function 3.2. The Schur-CUE connection 3.3. A plethystic-RKHS perspective on duality 3.4. CFKRS with RKHS 3.5. Duality in H N 4. Applications 4.1. The Keating-Snaith theorem 4.2. Autocorrelations of the characteristic polynomial in the microscopic setting 4.3. Ratios of the characteristic polynomial in the microscopic setting 4.4. The mid-secular coefficients 4.5. Back to the autocorrelations : the randomisation paradigm 4.6. The truncated characteristic polynomial 4.7. The Birkhoff polytope 4.8. Iterated derivatives of the characteristic polynomial 4.9. Sums of divisor functions in F q [X] 4.10. The moments of moments 5. Ultimate remarks 5.1. The Hankel form of the limit 5.2. Expansions 5.3. Other approaches to s λ [A] 5.4. Summary of the encountered functionals and limits 5.5. Partial conclusion and future work Appendix A. Probabilistic representations A.1. A symmetric function extension of Gegenbauer polynomials A.2. An integral representation of h n [A] A.3. Probabilistic representations of h (κ) c,∞ A.4. Integrability of h (κ) c,∞ A.5. Integrability in c A.6. The supersymmetric case Appendix B. Truncated ζ function in the microscopic scaling B.1. Reminders B.2. A phase transition Acknowledgements References

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Symmetric Function Theory and Unitary Invariant Ensembles

Cited by 6 publications

References 30 publications

Maxima of log-correlated fields: some recent developments

Maxima of log-correlated fields: some recent developments

On the moments of characteristic polynomials

A new approach to the characteristic polynomial of a random unitary matrix

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