4,016 research outputs found
Spectral statistics for unitary transfer matrices of binary graphs
Quantum graphs have recently been introduced as model systems to study the
spectral statistics of linear wave problems with chaotic classical limits. It
is proposed here to generalise this approach by considering arbitrary, directed
graphs with unitary transfer matrices. An exponentially increasing contribution
to the form factor is identified when performing a diagonal summation over
periodic orbit degeneracy classes. A special class of graphs, so-called binary
graphs, is studied in more detail. For these, the conditions for periodic orbit
pairs to be correlated (including correlations due to the unitarity of the
transfer matrix) can be given explicitly. Using combinatorial techniques it is
possible to perform the summation over correlated periodic orbit pair
contributions to the form factor for some low--dimensional cases. Gradual
convergence towards random matrix results is observed when increasing the
number of vertices of the binary graphs.Comment: 18 pages, 8 figure
Predicting specificity in bZIP coiled-coil protein interactions
We present a method for predicting protein-protein interactions mediated by the coiled-coil motif. When tested on interactions between nearly all human and yeast bZIP proteins, our method identifies 70% of strong interactions while maintaining that 92% of predictions are correct. Furthermore, cross-validation testing shows that including the bZIP experimental data significantly improves performance. Our method can be used to predict bZIP interactions in other genomes and is a promising approach for predicting coiled-coil interactions more generally
A new correlator in quantum spin chains
We propose a new correlator in one-dimensional quantum spin chains, the
Emptiness Formation Probability (EFP). This is a natural generalization
of the Emptiness Formation Probability (EFP), which is the probability that the
first spins of the chain are all aligned downwards. In the EFP we let
the spins in question be separated by sites. The usual EFP corresponds to
the special case when , and taking allows us to quantify non-local
correlations. We express the EFP for the anisotropic XY model in a
transverse magnetic field, a system with both critical and non-critical
regimes, in terms of a Toeplitz determinant. For the isotropic XY model we find
that the magnetic field induces an interesting length scale.Comment: 6 pages, 1 figur
Freezing Transition, Characteristic Polynomials of Random Matrices, and the Riemann Zeta-Function
We argue that the freezing transition scenario, previously explored in the
statistical mechanics of 1/f-noise random energy models, also determines the
value distribution of the maximum of the modulus of the characteristic
polynomials of large N x N random unitary (CUE) matrices. We postulate that our
results extend to the extreme values taken by the Riemann zeta-function zeta(s)
over sections of the critical line s=1/2+it of constant length and present the
results of numerical computations in support. Our main purpose is to draw
attention to possible connections between the statistical mechanics of random
energy landscapes, random matrix theory, and the theory of the Riemann zeta
function.Comment: published version with a few misprints corrected and references adde
On the resonance eigenstates of an open quantum baker map
We study the resonance eigenstates of a particular quantization of the open
baker map. For any admissible value of Planck's constant, the corresponding
quantum map is a subunitary matrix, and the nonzero component of its spectrum
is contained inside an annulus in the complex plane, . We consider semiclassical sequences of eigenstates, such that the
moduli of their eigenvalues converge to a fixed radius . We prove that, if
the moduli converge to , then the sequence of eigenstates
converges to a fixed phase space measure . The same holds for
sequences with eigenvalue moduli converging to , with a different
limit measure . Both these limiting measures are supported on
fractal sets, which are trapped sets of the classical dynamics. For a general
radius , we identify families of eigenstates with
precise self-similar properties.Comment: 32 pages, 2 figure
Applications and generalizations of Fisher-Hartwig asymptotics
Fisher-Hartwig asymptotics refers to the large form of a class of
Toeplitz determinants with singular generating functions. This class of
Toeplitz determinants occurs in the study of the spin-spin correlations for the
two-dimensional Ising model, and the ground state density matrix of the
impenetrable Bose gas, amongst other problems in mathematical physics. We give
a new application of the original Fisher-Hartwig formula to the asymptotic
decay of the Ising correlations above , while the study of the Bose gas
density matrix leads us to generalize the Fisher-Hartwig formula to the
asymptotic form of random matrix averages over the classical groups and the
Gaussian and Laguerre unitary matrix ensembles. Another viewpoint of our
generalizations is that they extend to Hankel determinants the Fisher-Hartwig
asymptotic form known for Toeplitz determinants.Comment: 25 page
Negative moments of characteristic polynomials of random GOE matrices and singularity-dominated strong fluctuations
We calculate the negative integer moments of the (regularized) characteristic
polynomials of N x N random matrices taken from the Gaussian Orthogonal
Ensemble (GOE) in the limit as . The results agree nontrivially
with a recent conjecture of Berry & Keating motivated by techniques developed
in the theory of singularity-dominated strong fluctuations. This is the first
example where nontrivial predictions obtained using these techniques have been
proved.Comment: 13 page
Characteristic Polynomials of Sample Covariance Matrices: The Non-Square Case
We consider the sample covariance matrices of large data matrices which have
i.i.d. complex matrix entries and which are non-square in the sense that the
difference between the number of rows and the number of columns tends to
infinity. We show that the second-order correlation function of the
characteristic polynomial of the sample covariance matrix is asymptotically
given by the sine kernel in the bulk of the spectrum and by the Airy kernel at
the edge of the spectrum. Similar results are given for real sample covariance
matrices
Tsunami Deposit Research: Fidelity of the Tsunami Record, Ephemeral Nature, Tsunami Deposits Characteristics, Remobilization of Sediment by Later Waves, and Boulder Movements
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