220,698 research outputs found
The forward approximation as a mean field approximation for the Anderson and Many Body Localization transitions
In this paper we analyze the predictions of the forward approximation in some
models which exhibit an Anderson (single-) or many-body localized phase. This
approximation, which consists in summing over the amplitudes of only the
shortest paths in the locator expansion, is known to over-estimate the critical
value of the disorder which determines the onset of the localized phase.
Nevertheless, the results provided by the approximation become more and more
accurate as the local coordination (dimensionality) of the graph, defined by
the hopping matrix, is made larger. In this sense, the forward approximation
can be regarded as a mean field theory for the Anderson transition in infinite
dimensions. The sum can be efficiently computed using transfer matrix
techniques, and the results are compared with the most precise exact
diagonalization results available.
For the Anderson problem, we find a critical value of the disorder which is
off the most precise available numerical value already in 5 spatial
dimensions, while for the many-body localized phase of the Heisenberg model
with random fields the critical disorder is strikingly close
to the most recent results obtained by exact diagonalization. In both cases we
obtain a critical exponent . In the Anderson case, the latter does not
show dependence on the dimensionality, as it is common within mean field
approximations.
We discuss the relevance of the correlations between the shortest paths for
both the single- and many-body problems, and comment on the connections of our
results with the problem of directed polymers in random medium
Fractal geometry of critical Potts clusters
Numerical simulations on the total mass, the numbers of bonds on the hull,
external perimeter, singly connected bonds and gates into large fjords of the
Fortuin-Kasteleyn clusters for two-dimensional q-state Potts models at
criticality are presented. The data are found consistent with the recently
derived corrections-to-scaling theory. However, the approach to the asymptotic
region is slow, and the present range of the data does not allow a unique
identification of the exact correction exponentsComment: 7 pages, 8 figures, Late
Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer
The number of solid partitions of a positive integer is an unsolved problem
in combinatorial number theory. In this paper, solid partitions are studied
numerically by the method of exact enumeration for integers up to 50 and by
Monte Carlo simulations using Wang-Landau sampling method for integers up to
8000. It is shown that, for large n, ln[p(n)]/n^(3/4) = 1.79 \pm 0.01, where
p(n) is the number of solid partitions of the integer n. This result strongly
suggests that the MacMahon conjecture for solid partitions, though not exact,
could still give the correct leading asymptotic behaviour.Comment: 6 pages, 4 figures, revtex
Fast and Slow Coherent Cascades in Anti-de Sitter Spacetime
We study the phase and amplitude dynamics of small perturbations in 3+1
dimensional Anti-de Sitter spacetime using the truncated resonant
approximation, also known as the Two Time Framework (TTF). We analyse the phase
spectrum for different classes of initial data and find that higher frequency
modes turn on with coherently aligned phases. Combining numerical and
analytical results, we conjecture that there is a class of initial conditions
that collapse in infinite slow time and to which the well-studied case of the
two-mode, equal energy initial data belongs. We additionally study
perturbations that collapse in finite time, and find that the energy spectrum
approaches a power law, with the energy per mode scaling approximately as the
inverse first power of the frequency.Comment: 19 pages, multiple figures. v2: version published in CQ
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