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

Abstract

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

Maxima of log-correlated fields: some recent developments

“…In order to capture the 'quasi-randomness' of the primes, Harper introduces the following random variables. Let (U p , p prime) be a sequence of independent random variables distributed uniformly on the unit circle 19 . Heuristically speaking 20 , U p models p −iτ , see also figure 6.…”

“…It is also interesting to remark that Barhoumi-Andréani [19] has recently proposed a general framework rooted in symmetric function theory for understanding various averages of unitary characteristic polynomials. Through this lens, one can rephrase and rederive many classical results in unitary random matrix theory.…”

“…One can analytically continue ζ(s) to all of the complex plane, with the exception of a simple pole at s = 1 (many proofs of this fact can be found, for example, in [72]). A consequence of the meromorphic continuation is the functional equation, (19) ζ(s) = 2 s π s−1 sin πs 2 Γ(1 − s)ζ(1 − s). Using (19), it is easy to see that there are zeros at the negative even integers coming from the sine function.…”

“…A consequence of the meromorphic continuation is the functional equation, (19) ζ(s) = 2 s π s−1 sin πs 2 Γ(1 − s)ζ(1 − s). Using (19), it is easy to see that there are zeros at the negative even integers coming from the sine function. These are known as trivial zeros.…”

“…Most broadly, an L-function is a Dirichlet series with an Euler product and a functional equation (cf. (17), (18), and (19)). In order to keep this review self-contained we focus on two canonical extensions.…”