38 research outputs found
Statistics of two-dimensional random walks, the "cyclic sieving phenomenon" and the Hofstadter model
We focus on the algebraic area probability distribution of planar random
walks on a square lattice with , , and steps right, left,
up and down. We aim, in particular, at the algebraic area generating function
evaluated at Q=e^{2\i\pi\over q}, a root of unity,
when both and are multiples of . In the simple case of
staircase walks, a geometrical interpretation of
in terms of the cyclic sieving phenomenon is
illustrated. Then, an expression for , which is
relevant to the Stembridge's case, is proposed. Finally, the related problem of
evaluating the n-th moments of the Hofstadter Hamiltonian in the commensurate
case is addressed.Comment: 13 pages, LaTeX 2
The third virial coefficient of anyons revisited
We use the method of solving the three-anyon problem developed in our earlier
publication to evaluate numerically the third virial coefficient of free
anyons. In order to improve precision, we explicitly correct for truncation
effects. The present calculation is about three orders of magnitude more
precise than the previous Monte Carlo calculation and indicates the presence of
a term with a very small coefficient .Comment: 10 pages, LATEX 2.09, 4 Postscript figures attached; explanations
adde
Anyon trajectories and the systematics of the three-anyon spectrum
We develop the concept of trajectories in anyon spectra, i.e., the continuous
dependence of energy levels on the kinetic angular momentum. It provides a more
economical and unified description, since each trajectory contains an infinite
number of points corresponding to the same statistics. For a system of
non-interacting anyons in a harmonic potential, each trajectory consists of two
infinite straight line segments, in general connected by a nonlinear piece. We
give the systematics of the three-anyon trajectories. The trajectories in
general cross each other at the bosonic/fermionic points. We use the
(semi-empirical) rule that all such crossings are true crossings, i.e.\ the
order of the trajectories with respect to energy is opposite to the left and to
the right of a crossing.Comment: 15 pages LaTeX + 1 attached uuencoded gzipped file with 7 figure
The Lowest Landau Level Anyon Equation of State in the Anti-screening Regime
The thermodynamics of the anyon model projected on the lowest Landau level
(LLL) of an external magnetic field is addressed in the anti-screening regime,
where the flux tubes carried by the anyons are parallel to the magnetic field.
It is claimed that the LLL-anyon equation of state, which is known in the
screening regime, can be analytically continued in the statistical parameter
across the Fermi point to the antiscreening regime up to the vicinity (whose
width tends to zero when the magnetic field becomes infinite) of the Bose
point. There, an unphysical discontinuity arises due to the dropping of the
non-LLL eigenstates which join the LLL, making the LLL approximation no longer
valid. However, taking into account the effect of the non-LLL states at the
Bose point would only smoothen the discontinuity and not alter the physics
which is captured by the LLL projection: Close to the Bose point, the critical
filling factor either goes to infinity (usual bosons) in the screening
situation, or to 1/2 in the anti-screening situation, the difference between
the flux tubes orientation being relevant even when they carry an infinitesimal
fraction of the flux quantum. An exclusion statistics interpretation is
adduced, which explains this situation in semiclassical terms. It is further
shown how the exact solutions of the 3-anyon problem support this scenario as
far as the third cluster coefficient is concerned.Comment: 14 pages, 3 figures, LaTex 2
Exact Multiplicities in the Three-Anyon Spectrum
Using the symmetry properties of the three-anyon spectrum, we obtain exactly
the multiplicities of states with given energy and angular momentum. The
results are shown to be in agreement with the proper quantum mechanical and
semiclassical considerations, and the unexplained points are indicated.Comment: 16 pages plus 3 postscript figures, Kiev Institute for Theoretical
Physics preprint ITP-93-32