Triple correlation of the Riemann zeros

We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested in 1996 by Bogomolny and Keating taking inspiration from semi-classical methods. At that point they did not write out the answer explicitly, so we do that here, illustrating that by our method all the lower order terms down to the constant can be calculated rigourously if one assumes the ratios conjecture of Conrey, Farmer and Zirnbauer. Bogomolny and Keating returned to their previous results simultaneously with this current work, and have written out the full expression. The result presented in this paper agrees precisely with their formula, as well as with our numerical computations, which we include here. We also include an alternate proof of the triple correlation of eigenvalues from random U(N) matrices which follows a nearly identical method to that for the Riemann zeros, but is based on the theorem for averages of ratios of characteristic polynomials

Developments in Random Matrix Theory

In this preface to the Journal of Physics A, Special Edition on Random Matrix Theory, we give a review of the main historical developments of random matrix theory. A short summary of the papers that appear in this special edition is also given.Comment: 22 pages, Late

Stochastic Models for Replication Origin Spacings in Eukaryotic DNA Replication

We consider eukaryotic DNA replication and in particular the role of replication origins in this process. We focus on origins which are `active' - that is, trigger themselves in the process before being read by the replication forks of other origins. We initially consider the spacings of these active replication origins in comparison to certain probability distributions of spacings taken from random matrix theory. We see how the spacings between neighbouring eigenvalues from certain collections of random matrices has some potential for modelling the spacing between active origins. This suitability can be further augmented with the use of uniform thinning which acts as a continuous deformation between correlated eigenvalue spacings and exponential (Poissonian) spacings. We model the process as a modified 2D Poisson process with an added exclusion rule to identify active points based on their position on the chromosome and trigger time relative to other origins. We see how this can be reduced to a stochastic geometry problem and show analytically that two active origins are unlikely to be close together, regardless of how many non-active points are between them. In particular, we see how these active origins repel linearly. We then see how data from various DNA datasets match with simulations from our model. We see that whilst there is variety in the DNA data, comparing the data with the model provides insight into the replication origin distribution of various organisms.Comment: 18 pages, 26 figure

Autocorrelation of Random Matrix Polynomials

We calculate the autocorrelation functions (or shifted moments) of the characteristic polynomials of matrices drawn uniformly with respect to Haar measure from the groups U(N), O(2N) and USp(2N). In each case the result can be expressed in three equivalent forms: as a determinant sum (and hence in terms of symmetric polynomials), as a combinatorial sum, and as a multiple contour integral. These formulae are analogous to those previously obtained for the Gaussian ensembles of Random Matrix Theory, but in this case are identities for any size of matrix, rather than large-matrix asymptotic approximations. They also mirror exactly autocorrelation formulae conjectured to hold for L-functions in a companion paper. This then provides further evidence in support of the connection between Random Matrix Theory and the theory of L-functions

Boundary conditions associated with the Painlev\'e III' and V evaluations of some random matrix averages

In a previous work a random matrix average for the Laguerre unitary ensemble, generalising the generating function for the probability that an interval $(0,s)$ at the hard edge contains $k$ eigenvalues, was evaluated in terms of a Painlev\'e V transcendent in $\sigma$-form. However the boundary conditions for the corresponding differential equation were not specified for the full parameter space. Here this task is accomplished in general, and the obtained functional form is compared against the most general small $s$ behaviour of the Painlev\'e V equation in $\sigma$-form known from the work of Jimbo. An analogous study is carried out for the the hard edge scaling limit of the random matrix average, which we have previously evaluated in terms of a Painlev\'e \IIId transcendent in $\sigma$-form. An application of the latter result is given to the rapid evaluation of a Hankel determinant appearing in a recent work of Conrey, Rubinstein and Snaith relating to the derivative of the Riemann zeta function

Multiplying unitary random matrices - universality and spectral properties

In this paper we calculate, in the large N limit, the eigenvalue density of an infinite product of random unitary matrices, each of them generated by a random hermitian matrix. This is equivalent to solving unitary diffusion generated by a hamiltonian random in time. We find that the result is universal and depends only on the second moment of the generator of the stochastic evolution. We find indications of critical behavior (eigenvalue spacing scaling like $1/N^{3/4}$) close to $\theta=\pi$ for a specific critical evolution time $t_c$.Comment: 12 pages, 2 figure

The lowest eigenvalue of Jacobi random matrix ensembles and Painlev\'e VI

We present two complementary methods, each applicable in a different range, to evaluate the distribution of the lowest eigenvalue of random matrices in a Jacobi ensemble. The first method solves an associated Painleve VI nonlinear differential equation numerically, with suitable initial conditions that we determine. The second method proceeds via constructing the power-series expansion of the Painleve VI function. Our results are applied in a forthcoming paper in which we model the distribution of the first zero above the central point of elliptic curve L-function families of finite conductor and of conjecturally orthogonal symmetry.Comment: 30 pages, 2 figure

Quantum gray solitons in confining potentials

We define and study hole-like excitations (the Lieb II mode) in a weakly interacting Bose liquid subject to external confinement. These excitations are obtained by semiclassical quantization of gray solitons propagating on top of a Thomas-Fermi background. Radiation of phonons by an accelerated gray soliton leads to a finite life-time for the trapped Lieb II mode. It is shown that, for a large number of trapped atoms, most of the Lieb II levels can be experimentally resolved.Comment: 5 pages, 2 figure