814 research outputs found
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
Correlations of eigenvalues and Riemann zeros
We present a new approach to obtaining the lower order terms for
-correlation of the zeros of the Riemann zeta function. Our approach is
based on the `ratios conjecture' of Conrey, Farmer, and Zirnbauer. Assuming the
ratios conjecture we prove a formula which explicitly gives all of the lower
order terms in any order correlation. Our method works equally well for random
matrix theory and gives a new expression, which is structurally the same as
that for the zeta function, for the -correlation of eigenvalues of matrices
from U(N)
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 at the hard edge contains eigenvalues, was evaluated in terms of
a Painlev\'e V transcendent in -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 behaviour of
the Painlev\'e V equation in -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 -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
On the spacing distribution of the Riemann zeros: corrections to the asymptotic result
It has been conjectured that the statistical properties of zeros of the
Riemann zeta function near z = 1/2 + \ui E tend, as , to the
distribution of eigenvalues of large random matrices from the Unitary Ensemble.
At finite numerical results show that the nearest-neighbour spacing
distribution presents deviations with respect to the conjectured asymptotic
form. We give here arguments indicating that to leading order these deviations
are the same as those of unitary random matrices of finite dimension , where is a well
defined constant.Comment: 9 pages, 3 figure
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 ) close to for a specific critical evolution time
.Comment: 12 pages, 2 figure
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