587 research outputs found

### Winding number correlation for a Brownian loop in a plane

A Brownian loop is a random walk circuit of infinitely many, suitably
infinitesimal, steps. In a plane such a loop may or may not enclose a marked
point, the origin, say. If it does so it may wind arbitrarily many times,
positive or negative, around that point. Indeed from the (long known)
probability distribution, the mean square winding number is infinite, so all
statistical moments - averages of powers of the winding number - are infinity
(even powers) or zero (odd powers, by symmetry). If an additional marked point
is introduced at some distance from the origin, there are now two winding
numbers, which are correlated. That correlation, the average of the product of
the two winding numbers, is finite and is calculated here. The result takes the
form of a single well-convergent integral that depends on a single parameter -
the suitably scaled separation of the marked points. The integrals of the
correlation weighted by powers of the separation are simple factorial
expressions. Explicit limits of the correlation for small and large separation
of the marked points are found.Comment: The right hand sides of various equations were missing factors of 1/2
or 1/4, now correcte

### An experiment on the shifts of reflected C-lines

An experiment is described that tests theoretical predictions on how C-lines
incident obliquely on a surface behave on reflection. C-lines in a polarised
wave are the analogues of the optical vortices carried by a complex scalar
wave, which is the usual model for describing light and other electromagnetic
waves. The centre of a laser beam that carries a (degenerate) C-line is shifted
on reflection by the well-known Goos-H\"anchen and Imbert-Fedorov effects, but
the C-line itself splits into two, both of which are shifted longitudinally and
laterally; their shifts are different from that of the beam centre. To maximise
the effect to be measured, internal reflection in a glass prism close to the
critical angle was used. In a simple situation like this two recently published
independent theories of C-line reflection overlap and it is shown that their
predictions are identical. The measured differences in the lateral shifts of
the two reflected C-lines are compared with theoretical expectations over a
range of incidence angles.Comment: 9 pages, 2 figure

### 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 $N \to \infty$. 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

### Fluctuations and Ergodicity of the Form Factor of Quantum Propagators and Random Unitary Matrices

We consider the spectral form factor of random unitary matrices as well as of
Floquet matrices of kicked tops. For a typical matrix the time dependence of
the form factor looks erratic; only after a local time average over a suitably
large time window does a systematic time dependence become manifest. For
matrices drawn from the circular unitary ensemble we prove ergodicity: In the
limits of large matrix dimension and large time window the local time average
has vanishingly small ensemble fluctuations and may be identified with the
ensemble average. By numerically diagonalizing Floquet matrices of kicked tops
with a globally chaotic classical limit we find the same ergodicity. As a
byproduct we find that the traces of random matrices from the circular
ensembles behave very much like independent Gaussian random numbers. Again,
Floquet matrices of chaotic tops share that universal behavior. It becomes
clear that the form factor of chaotic dynamical systems can be fully faithful
to random-matrix theory, not only in its locally time-averaged systematic time
dependence but also in its fluctuations.Comment: 12 pages, RevTEX, 4 figures in eps forma

### Two-point correlations of the Gaussian symplectic ensemble from periodic orbits

We determine the asymptotics of the two-point correlation function for
quantum systems with half-integer spin which show chaotic behaviour in the
classical limit using a method introduced by Bogomolny and Keating [Phys. Rev.
Lett. 77 (1996) 1472-1475]. For time-reversal invariant systems we obtain the
leading terms of the two-point correlation function of the Gaussian symplectic
ensemble. Special attention has to be paid to the role of Kramers' degeneracy.Comment: 7 pages, no figure

### Geometrically Induced Gauge Structure on Manifolds Embedded in a Higher Dimensional Space

We explain in a context different from that of Maraner the formalism for
describing motion of a particle, under the influence of a confining potential,
in a neighbourhood of an n-dimensional curved manifold M^n embedded in a
p-dimensional Euclidean space R^p with p >= n+2. The effective Hamiltonian on
M^n has a (generally non-Abelian) gauge structure determined by geometry of
M^n. Such a gauge term is defined in terms of the vectors normal to M^n, and
its connection is called the N-connection. In order to see the global effect of
this type of connections, the case of M^1 embedded in R^3 is examined, where
the relation of an integral of the gauge potential of the N-connection (i.e.,
the torsion) along a path in M^1 to the Berry's phase is given through Gauss
mapping of the vector tangent to M^1. Through the same mapping in the case of
M^1 embedded in R^p, where the normal and the tangent quantities are exchanged,
the relation of the N-connection to the induced gauge potential on the
(p-1)-dimensional sphere S^{p-1} (p >= 3) found by Ohnuki and Kitakado is
concretely established. Further, this latter which has the monopole-like
structure is also proved to be gauge-equivalent to the spin-connection of
S^{p-1}. Finally, by extending formally the fundamental equations for M^n to
infinite dimensional case, the present formalism is applied to the field theory
that admits a soliton solution. The resultant expression is in some respects
different from that of Gervais and Jevicki.Comment: 52 pages, PHYZZX. To be published in Int. J. Mod. Phys.

### Shrunk loop theorem for the topology probabilities of closed Brownian (or Feynman) paths on the twice punctured plane

The shrunk loop theorem presented here is an integral identity which
facilitates the calculation of the relative probability (or probability
amplitude) of any given topology that a free, closed Brownian or Feynman path
of a given 'duration' might have on the twice punctured plane (the plane with
two marked points). The result is expressed as a scattering series of integrals
of increasing dimensionality based on the maximally shrunk version of the path.
Physically, this applies in different contexts: (i) the topology probability of
a closed ideal polymer chain on a plane with two impassable points, (ii) the
trace of the Schroedinger Green function, and thence spectral information, in
the presence of two Aharonov-Bohm fluxes, (iii) the same with two branch points
of a Riemann surface instead of fluxes. Our theorem starts with the Stovicek
expansion for the Green function in the presence of two Aharonov-Bohm flux
lines, which itself is based on the famous Sommerfeld one puncture point
solution of 1896 (the one puncture case has much easier topology, just one
winding number). Stovicek's expansion itself can supply the results at the
expense of choosing a base point on the loop and then integrating it away. The
shrunk loop theorem eliminates this extra two dimensional integration,
distilling the topology from the geometry.Comment: 29 pages, 5 figures (accepted by J. Phys. A: Math. Gen.

### Semiclassical Approach to Parametric Spectral Correlation with Spin 1/2

The spectral correlation of a chaotic system with spin 1/2 is universally
described by the GSE (Gaussian Symplectic Ensemble) of random matrices in the
semiclassical limit. In semiclassical theory, the spectral form factor is
expressed in terms of the periodic orbits and the spin state is simulated by
the uniform distribution on a sphere. In this paper, instead of the uniform
distribution, we introduce Brownian motion on a sphere to yield the parametric
motion of the energy levels. As a result, the small time expansion of the form
factor is obtained and found to be in agreement with the prediction of
parametric random matrices in the transition within the GSE universality class.
Moreover, by starting the Brownian motion from a point distribution on the
sphere, we gradually increase the effect of the spin and calculate the form
factor describing the transition from the GOE (Gaussian Orthogonal Ensemble)
class to the GSE class.Comment: 25 pages, 2 figure

### Notes on Conformal Invisibility Devices

As a consequence of the wave nature of light, invisibility devices based on
isotropic media cannot be perfect. The principal distortions of invisibility
are due to reflections and time delays. Reflections can be made exponentially
small for devices that are large in comparison with the wavelength of light.
Time delays are unavoidable and will result in wave-front dislocations. This
paper considers invisibility devices based on optical conformal mapping. The
paper shows that the time delays do not depend on the directions and impact
parameters of incident light rays, although the refractive-index profile of any
conformal invisibility device is necessarily asymmetric. The distortions of
images are thus uniform, which reduces the risk of detection. The paper also
shows how the ideas of invisibility devices are connected to the transmutation
of force, the stereographic projection and Escheresque tilings of the plane

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