Jon P Keating

Integral moments of L -functions

We give a new heuristic for all of the main terms in the integral moments of various families of primitive

The evolution of bubble size distributions in volcanic eruptions

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Inferring volcanic degassing processes from vesicle size distributions

-functions. The results agree with previous conjectures for the leading order terms. Our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary, orthogonal, or symplectic matrices, where the moments are defined by the appropriate group averages. This lends support to the idea that arithmetical

A rule for quantizing chaos

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Random matrix theory and the Riemann zeros II: n-point correlations

-functions have a spectral interpretation, and that their value distributions can be modelled using Random Matrix Theory. Numerical examples show good agreement with our conjectures.

Random matrix theory and the derivative of the Riemann zeta function

We review observations of bubble size distributions (BSDs) generated during explosive volcanic eruptions and laboratory explosions, as inferred from vesicle size distributions found in the end products. Unimodal, polymodal, exponential and power law BSDs are common, even in the absence of coalescence, and both power law and exponential distributions have been generated in the same eruption. To date theoretical models have proposed incompatible mechanisms for producing the various distributions. We here present a unifying mechanism. Data from our laboratory analogue experiments suggest that power law distributions are associated with highly non-equilibrium degassing. A numerical model is developed in which bubbles nucleate repeatedly and grow in the spaces between those of previous generations, where, in a non-equilibrium degassing scenario, the volatile concentration remains high. This process causes the BSD to evolve from unimodal, through exponential, into a power law. The exponent of the power law is a measure of the number of nucleation events, or the duration of the nucleation period compared with the timescale of bubble growth. The mathematical inevitability of the evolution from unimodal (Poissonian) to power law is discussed. The findings may resolve the apparent contradiction between the equilibrium degassing conduit flow models and the non-equilibrium degassing conditions derived from bubble growth models of explosive volcanic eruptions. The process of ongoing nucleation is the mechanism whereby the volcanic system maintains near-equilibrium in the case of rapid depressurisation and slow volatile diffusion.

Random matrices and L-functions

Both power law and exponential vesicle size distributions (VSDs) have been observed in many different types of volcanic rocks. We present results of computer simulations and laboratory analogue experiments which reproduce these findings and show that the distributions can be interpreted as the product of continuous bubble nucleation resulting from non-equilibrium degassing. This ongoing nucleation causes the bubbles to evolve through an exponential size distribution into a power law size distribution as nucleation and growth progress. These findings may explain the apparent contradiction between present models of bubble growth in magmas, which predict that degassing in explosive eruptions is a non-equilibrium process, and models of conduit flow, which assume perfect equilibrium degassing. The process of continuous nucleation is the mechanism whereby the volcanic system maintains near-equilibrium in the case of rapid depressurization and slow volatile diffusion.

Random matrix theory and the Riemann zeros. I. Three- and four-point correlations

The authors find a real function Delta (E) whose zeros approximate the quantum energy levels of a system with chaotic classical trajectories. Delta (E) is a finite sum over combinations of classical periodic orbits. It is obtained from Gutzwillers infinite and divergent sum (1982), representing the spectral density in terms of periodic orbits, by means of a resummation conjectured by analogy with a derivation of the Riemann-Siegel formula for the Riemann zeros. They assess the practicality of the quantization condition.

Autocorrelation of random matrix polynomials

Montgomery has conjectured that the non-trivial zeros of the Riemann zeta-function are pairwise distributed like the eigenvalues of matrices in the Gaussian unitary ensemble (GUE) of random matrix theory (RMT). In this respect, they provide an important model for the statistical properties of the energy levels of quantum systems whose classical limits are strongly chaotic. We generalize this connection by showing that for all the n-point correlation function of the zeros is equivalent to the corresponding GUE result in the appropriate asymptotic limit. Our approach is based on previous demonstrations for the particular cases n = 2, 3, 4. It relies on several new combinatorial techniques, first for evaluating the multiple prime sums involved using a Hardy - Littlewood prime-correlation conjecture, and second for expanding the GUE correlation-function determinant. This constitutes the first complete demonstration of RMT behaviour for all orders of correlation in a simple, deterministic model.

Asymptotic properties of the periodic orbits of the cat maps

Random matrix theory is used to model the asymptotics of the discrete moments of the derivative of the Riemann zeta function, ζ(s), evaluated at the complex zeros ½; + iγn. We also discuss the probability distribution of ln |ζ′(1/2 + iγn)|, proving the central limit theorem for the corresponding random matrix distribution and analysing its large deviations.