Finite size corrections in random matrix theory and Odlyzko's data set for the Riemann zeros

Abstract

Odlyzko has computed a data set listing more than $10^9$ successive Riemann zeros, starting at a zero number beyond $10^{23}$. The data set relates to random matrix theory since, according to the Montgomery-Odlyzko law, the statistical properties of the large Riemann zeros agree with the statistical properties of the eigenvalues of large random Hermitian matrices. Moreover, Keating and Snaith, and then Bogomolny and collaborators, have used $N\times N$ random unitary matrices to analyse deviations from this law. We contribute to this line of study in two ways. First, we point out that a natural process to apply to the data set is to thin it by deleting each member independently with some specified probability, and we proceed to compute empirical two-point correlation functions and nearest neighbour spacings in this setting. Second, we show how to characterise the order $1/N^2$ correction term to the spacing distribution for random unitary matrices in terms of a second order differential equation with coefficients that are Painlevé transcendents, and where the thinning parameter appears only in the boundary condition. This equation can be solved numerically using a power series method. Comparison with the Riemann zero data shows accurate agreement.

22 pages, 10 figures, Version 2 added some new references in bibliography, Version 3 corrected the scaling on the spacing distribution and some typos

References in corpus (6)

• On the asymptotic behavior of a log gas in the bulk scaling limit in the presence of a varying external potential I
• On the spacing distribution of the Riemann zeros: corrections to the asymptotic result
• Riemann zeta function and quantum chaos
• Random Matrix Theory and the Sixth Painlevé Equation
• A note on the expansion of the smallest eigenvalue distribution of the LUE at the hard edge
• Spacings - An Example for Universality in Random Matrix Theory

Cited by in corpus (7)

• Large deformations of the Tracy-Widom distribution I. Non-oscillatory asymptotics
• Power Spectrum of Long Eigenlevel Sequences in Quantum Chaotic Systems
• Optimal soft edge scaling variables for the Gaussian and Laguerre even $\beta$ ensembles
• Functional form for the leading correction to the distribution of the largest eigenvalue in the GUE and LUE
• Leading corrections to the scaling function on the diagonal for the two-dimensional Ising model
• Distributions of consecutive level spacings of Gaussian unitary ensemble and their ratio: ab initio derivation
• Power spectra and autocovariances of level spacings beyond the Dyson conjecture