Exchangeable arrays and integrable systems for characteristic polynomials of random matrices

• DOI: 10.48550/arxiv.2407.19233
• Publication Date: 2024-07-27
• Authors: Theodoros Assiotis; Mustafa Alper Gunes; Jonathan P. Keating; Fei Wei
• Keywords: Mathematics - Number Theory; Probability (math.PR); FOS: Mathematics; FOS: Physical sciences; Mathematical Physics (math-ph); Number Theory (math.NT); Mathematics - Probability; Mathematical Physics

The joint moments of the derivatives characteristic polynomial a random unitary matrix, and also variant that is real on unit circle, in large matrix size limit, have been studied intensively past twenty five years, partly relation to conjectural connections Riemann zeta-function Hardy's function. We completely settle most general version problem convergence these moments, after they are suitably rescaled, for an arbitrary number with positive exponents. Our approach relies hidden, higher-order exchangeable structure, array. Using probabilistic techniques, we then give combinatorial formula leading order coefficient asymptotics when power itself exponents integers, terms finite finite-dimensional integrals which explicitly computable. Finally, develop method, based class Hankel determinants shifted by partitions, allows us exact representation all size, Painlev\'e V transcendents, large-matrix limit $\sigma$-Painlev\'e III' equation. This gives efficient way compute coefficients explicitly. methods can be used obtain analogous results other models sharing same features.