9,043 research outputs found
Field Theory And Second Renormalization Group For Multifractals In Percolation
The field-theory for multifractals in percolation is reformulated in such a
way that multifractal exponents clearly appear as eigenvalues of a second
renormalization group. The first renormalization group describes geometrical
properties of percolation clusters, while the second-one describes electrical
properties, including noise cumulants. In this context, multifractal exponents
are associated with symmetry-breaking fields in replica space. This provides an
explanation for their observability. It is suggested that multifractal
exponents are ''dominant'' instead of ''relevant'' since there exists an
arbitrary scale factor which can change their sign from positive to negative
without changing the Physics of the problem.Comment: RevTex, 10 page
Random Geometric Graphs
We analyse graphs in which each vertex is assigned random coordinates in a
geometric space of arbitrary dimensionality and only edges between adjacent
points are present. The critical connectivity is found numerically by examining
the size of the largest cluster. We derive an analytical expression for the
cluster coefficient which shows that the graphs are distinctly different from
standard random graphs, even for infinite dimensionality. Insights relevant for
graph bi-partitioning are included.Comment: 16 pages, 10 figures. Minor changes. Added reference
Transfer-matrix approach to the three-dimensional bond percolation: An application of Novotny's formalism
A transfer-matrix simulation scheme for the three-dimensional (d=3) bond
percolation is presented. Our scheme is based on Novotny's transfer-matrix
formalism, which enables us to consider arbitrary (integral) number of sites N
constituting a unit of the transfer-matrix slice even for d=3. Such an
arbitrariness allows us to perform systematic finite-size-scaling analysis of
the criticality at the percolation threshold. Diagonalizing the transfer matrix
for N =4,5,...,10, we obtain an estimate for the correlation-length critical
exponent nu = 0.81(5)
The dimension of the range of a transient random walk
We find formulas for the macroscopic Minkowski and Hausdorff dimensions of
the range of an arbitrary transient walk in Z^d. This endeavor solves a problem
of Barlow and Taylor (1991).Comment: 37 pages, 5 figure
Dynamics around the Site Percolation Threshold on High-Dimensional Hypercubic Lattices
Recent advances on the glass problem motivate reexamining classical models of
percolation. Here, we consider the displacement of an ant in a labyrinth near
the percolation threshold on cubic lattices both below and above the upper
critical dimension of simple percolation, d_u=6. Using theory and simulations,
we consider the scaling regime part, and obtain that both caging and
subdiffusion scale logarithmically for d >= d_u. The theoretical derivation
considers Bethe lattices with generalized connectivity and a random graph
model, and employs a scaling analysis to confirm that logarithmic scalings
should persist in the infinite dimension limit. The computational validation
employs accelerated random walk simulations with a transfer-matrix description
of diffusion to evaluate directly the dynamical critical exponents below d_u as
well as their logarithmic scaling above d_u. Our numerical results improve
various earlier estimates and are fully consistent with our theoretical
predictions.Comment: 12 pages, 6 figure
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