19 research outputs found
Finite time and asymptotic behaviour of the maximal excursion of a random walk
We evaluate the limit distribution of the maximal excursion of a random walk
in any dimension for homogeneous environments and for self-similar supports
under the assumption of spherical symmetry. This distribution is obtained in
closed form and is an approximation of the exact distribution comparable to
that obtained by real space renormalization methods. Then we focus on the early
time behaviour of this quantity. The instantaneous diffusion exponent
exhibits a systematic overshooting of the long time exponent. Exact results are
obtained in one dimension up to third order in . In two dimensions,
on a regular lattice and on the Sierpi\'nski gasket we find numerically that
the analytic scaling holds.Comment: 9 pages, 4 figures, accepted J. Phys.
Enumeration of simple random walks and tridiagonal matrices
We present some old and new results in the enumeration of random walks in one
dimension, mostly developed in works of enumerative combinatorics. The relation
between the trace of the -th power of a tridiagonal matrix and the
enumeration of weighted paths of steps allows an easier combinatorial
enumeration of the paths. It also seems promising for the theory of tridiagonal
random matrices .Comment: several ref.and comments added, misprints correcte
Schwinger-boson approach to quantum spin systems: Gaussian fluctuactions in the "natural" gauge
We compute the Gaussian-fluctuation corrections to the saddle-point
Schwinger-boson results using collective coordinate methods. Concrete
application to investigate the frustrated J1-J2 antiferromagnet on the square
lattice shows that, unlike the saddle-point predictions, there is a quantum
nonmagnetic phase for 0.53 < J2/J1 < 0.64. This result is obtained by
considering the corrections to the spin stiffness on large lattices and
extrapolating to the thermodynamic limit, which avoids the infinite-lattice
infrared divergencies associated to Bose condensation. The very good agreement
of our results with exact numerical values on finite clusters lends support to
the calculational scheme employed.Comment: 4 pages, Latex, 3 figures included as eps files,minor correction
Superconductivity and Quantum Spin Disorder in Cuprates
A fundamental connection between superconductivity and quantum spin
fluctuations in underdoped cuprates, is revealed. A variational calculation
shows that {\em Cooper pair hopping} strongly reduces the local magnetization
. This effect pertains to recent neutron scattering and muon spin rotation
measurements in which varies weakly with hole doping in the poorly
conducting regime, but drops precipitously above the onset of
superconductivity
The Expansion and Spin Correlations in Constrained Wavefunctions
We develop a large-N expansion for Gutzwiller projected spin states. We
consider valence bonds singlets, constructed by Schwinger bosons or fermions,
which are variational ground states for quantum antiferromagnets. This
expansion is simpler than the familiar expansions of the quantum Heisenberg
model, and thus more instructive. The diagrammatic rules of this expansion
allow us to prove certain identities to all orders in 1/N. We derive the
on-site spin fluctuations sum rule for arbitrary N. We calculate the
correlations of the one dimensional Valence Bonds Solid states and the
Gutzwiller Projected Fermi Gas upto order 1/N. For the bosons case, we are
surprised to find that the mean field, the order 1/N and the exact correlations
are simply proportional. For the fermions case, the 1/N correction enhances the
zone edge singularity. The comparison of our leading order terms to known
results for N=2, enhances our understanding of large-N approximations in
general.Comment: 36 pages, LaTe