47,206 research outputs found
On the Number of Incipient Spanning Clusters
In critical percolation models, in a large cube there will typically be more
than one cluster of comparable diameter. In 2D, the probability of
spanning clusters is of the order . In dimensions d>6, when
the spanning clusters proliferate: for the spanning
probability tends to one, and there typically are spanning
clusters of size comparable to |\C_{max}| \approx L^4. The rigorous results
confirm a generally accepted picture for d>6, but also correct some
misconceptions concerning the uniqueness of the dominant cluster. We
distinguish between two related concepts: the Incipient Infinite Cluster, which
is unique partly due to its construction, and the Incipient Spanning Clusters,
which are not. The scaling limits of the ISC show interesting differences
between low (d=2) and high dimensions. In the latter case (d>6 ?) we find
indication that the double limit: infinite volume and zero lattice spacing,
when properly defined would exhibit both percolation at the critical state and
infinitely many infinite clusters.Comment: Latex(2e), 42 p, 5 figures; to appear in Nucl. Phys. B [FS
Partition function zeros at first-order phase transitions: Pirogov-Sinai theory
This paper is a continuation of our previous analysis [BBCKK] of partition
functions zeros in models with first-order phase transitions and periodic
boundary conditions. Here it is shown that the assumptions under which the
results of [BBCKK] were established are satisfied by a large class of lattice
models. These models are characterized by two basic properties: The existence
of only a finite number of ground states and the availability of an appropriate
contour representation. This setting includes, for instance, the Ising, Potts
and Blume-Capel models at low temperatures. The combined results of [BBCKK] and
the present paper provide complete control of the zeros of the partition
function with periodic boundary conditions for all models in the above class.Comment: 46 pages, 2 figs; continuation of math-ph/0304007 and
math-ph/0004003, to appear in J. Statist. Phys. (special issue dedicated to
Elliott Lieb
Winterbottom Construction for Finite Range Ferromagnetic Models: An L_1 Approach
We provide a rigorous microscopic derivation of the thermodynamic description
of equilibrium crystal shapes in the presence of a substrate, first studied by
Winterbottom. We consider finite range ferromagnetic Ising models with pair
interactions in dimensions greater or equal to 3, and model the substrate by a
finite-range boundary magnetic field acting on the spins close to the bottom
wall of the box
Missing data in multiplex networks: a preliminary study
A basic problem in the analysis of social networks is missing data. When a
network model does not accurately capture all the actors or relationships in
the social system under study, measures computed on the network and ultimately
the final outcomes of the analysis can be severely distorted. For this reason,
researchers in social network analysis have characterised the impact of
different types of missing data on existing network measures. Recently a lot of
attention has been devoted to the study of multiple-network systems, e.g.,
multiplex networks. In these systems missing data has an even more significant
impact on the outcomes of the analyses. However, to the best of our knowledge,
no study has focused on this problem yet. This work is a first step in the
direction of understanding the impact of missing data in multiple networks. We
first discuss the main reasons for missingness in these systems, then we
explore the relation between various types of missing information and their
effect on network properties. We provide initial experimental evidence based on
both real and synthetic data.Comment: 7 page
Low-density series expansions for directed percolation I: A new efficient algorithm with applications to the square lattice
A new algorithm for the derivation of low-density series for percolation on
directed lattices is introduced and applied to the square lattice bond and site
problems. Numerical evidence shows that the computational complexity grows
exponentially, but with a growth factor \lambda < \protect{\sqrt[8]{2}},
which is much smaller than the growth factor \lambda = \protect{\sqrt[4]{2}}
of the previous best algorithm. For bond (site) percolation on the directed
square lattice the series has been extended to order 171 (158). Analysis of the
series yields sharper estimates of the critical points and exponents.Comment: 20 pages, 8 figures (3 of them > 1Mb
Low-density series expansions for directed percolation I: A new efficient algorithm with applications to the square lattice
A new algorithm for the derivation of low-density series for percolation on
directed lattices is introduced and applied to the square lattice bond and site
problems. Numerical evidence shows that the computational complexity grows
exponentially, but with a growth factor \lambda < \protect{\sqrt[8]{2}},
which is much smaller than the growth factor \lambda = \protect{\sqrt[4]{2}}
of the previous best algorithm. For bond (site) percolation on the directed
square lattice the series has been extended to order 171 (158). Analysis of the
series yields sharper estimates of the critical points and exponents.Comment: 20 pages, 8 figures (3 of them > 1Mb
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