1,596 research outputs found
Gaussian model of explosive percolation in three and higher dimensions
The Gaussian model of discontinuous percolation, recently introduced by
Ara\'ujo and Herrmann [Phys. Rev. Lett., 105, 035701 (2010)], is numerically
investigated in three dimensions, disclosing a discontinuous transition. For
the simple-cubic lattice, in the thermodynamic limit, we report a finite jump
of the order parameter, . The largest cluster at the
threshold is compact, but its external perimeter is fractal with fractal
dimension . The study is extended to hypercubic lattices up
to six dimensions and to the mean-field limit (infinite dimension). We find
that, in all considered dimensions, the percolation transition is
discontinuous. The value of the jump in the order parameter, the maximum of the
second moment, and the percolation threshold are analyzed, revealing
interesting features of the transition and corroborating its discontinuous
nature in all considered dimensions. We also show that the fractal dimension of
the external perimeter, for any dimension, is consistent with the one from
bridge percolation and establish a lower bound for the percolation threshold of
discontinuous models with finite number of clusters at the threshold
Recent advances and open challenges in percolation
Percolation is the paradigm for random connectivity and has been one of the
most applied statistical models. With simple geometrical rules a transition is
obtained which is related to magnetic models. This transition is, in all
dimensions, one of the most robust continuous transitions known. We present a
very brief overview of more than 60 years of work in this area and discuss
several open questions for a variety of models, including classical, explosive,
invasion, bootstrap, and correlated percolation
Percolation with long-range correlated disorder
Long-range power-law correlated percolation is investigated using Monte Carlo
simulations. We obtain several static and dynamic critical exponents as
function of the Hurst exponent which characterizes the degree of spatial
correlation among the occupation of sites. In particular, we study the fractal
dimension of the largest cluster and the scaling behavior of the second moment
of the cluster size distribution, as well as the complete and accessible
perimeters of the largest cluster. Concerning the inner structure and transport
properties of the largest cluster, we analyze its shortest path, backbone, red
sites, and conductivity. Finally, bridge site growth is also considered. We
propose expressions for the functional dependence of the critical exponents on
Watersheds are Schramm-Loewner Evolution curves
We show that in the continuum limit watersheds dividing drainage basins are
Schramm-Loewner Evolution (SLE) curves, being described by one single parameter
. Several numerical evaluations are applied to ascertain this. All
calculations are consistent with SLE, with ,
being the only known physical example of an SLE with . This lies
outside the well-known duality conjecture, bringing up new questions regarding
the existence and reversibility of dual models. Furthermore it constitutes a
strong indication for conformal invariance in random landscapes and suggests
that watersheds likely correspond to a logarithmic Conformal Field Theory (CFT)
with central charge .Comment: 5 pages and 4 figure
Watersheds in disordered media
This is the final published version. It first appeared at http://journal.frontiersin.org/article/10.3389/fphy.2015.00005/full#h11.What is the best way to divide a rugged landscape? Since ancient times, watersheds
separating adjacent water systems that flow, for example, toward different seas, have
been used to delimit boundaries. Interestingly, serious and even tense border disputes
between countries have relied on the subtle geometrical properties of these tortuous
lines. For instance, slight and even anthropogenic modifications of landscapes can produce
large changes in a watershed, and the effects can be highly nonlocal. Although the
watershed concept arises naturally in geomorphology, where it plays a fundamental role
in water management, landslide, and flood prevention, it also has important applications
in seemingly unrelated fields such as image processing and medicine. Despite the
far-reaching consequences of the scaling properties on watershed-related hydrological and
political issues, it was only recently that a more profound and revealing connection has
been disclosed between the concept of watershed and statistical physics of disordered
systems. This review initially surveys the origin and definition of a watershed line in a
geomorphological framework to subsequently introduce its basic geometrical and physical
properties. Results on statistical properties of watersheds obtained from artificial model
landscapes generated with long-range correlations are presented and shown to be in good
qualitative and quantitative agreement with real landscapes.We acknowledge financial support from the European Research Council (ERC) Advanced Grant 319968-FlowCCS, the Brazilian Agencies CNPq, CAPES, FUNCAP and FINEP, the FUNCAP/CNPq Pronex grant, the National Institute of Science and Technology for Complex Systems in Brazil, the Portuguese Foundation for Science and Technology (FCT) under contracts no. IF/00255/2013, PEst-OE/FIS/UI0618/2014, and EXCL/FIS-NAN/0083/2012, and the Swiss National Science Foundation under Grant No. P2EZP2-152188
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