Related papers: Geometrical clusters in two-dimensional random-fie…
We consider the behaviour of a critical system in the presence of a gradient perturbation of the couplings. In the direction of the gradient an interface region separates the ordered phase from the disordered one. We develop a scaling…
Two distinct distribution functions $P_{sp}(m)$ and $P_{ns}(m)$ of the scaled largest cluster sizes $m$ are obtained at the percolation threshold by numerical simulations, depending on the condition whether the lattice is actually spanned…
In the neighbourhood of the critical point, the correlation length of the spin-spin correlation function of the two-dimensional Ising model diverges. The correlation function permits a scaling limit in which the separation $N$ between spins…
We present a numerical study of topological descriptors of initially Gaussian and scale-free density perturbations evolving via gravitational instability in an expanding universe. We carefully evaluate and avoid numerical contamination in…
We consider long-range Bernoulli bond percolation on the $d$-dimensional hierarchical lattice in which each pair of points $x$ and $y$ are connected by an edge with probability $1-\exp(-\beta\|x-y\|^{-d-\alpha})$, where $0<\alpha<d$ is…
We devise a geometric description of bounded systems at criticality in any dimension $d$. This is achieved by altering the flat metric with a space dependent scale factor $\gamma(x)$, $x$ belonging to a general bounded domain $\Omega$.…
A scaling theory is used to derive the dependence of the average number <k> of spanning clusters at threshold on the lattice size L. This number should become independent of L for dimensions d<6, and vary as log L at d=6. The predictions…
The geometric mean is shown to be an appropriate statistic for the scale of a heavy-tailed coupled Gaussian distribution or equivalently the Student's t distribution. The coupled Gaussian is a member of a family of distributions…
Percolation models with multiple percolating clusters have attracted much attention in recent years. Here we use Monte Carlo simulations to study bond percolation on $L_{1}\times L_{2}$ planar random lattices, duals of random lattices, and…
The probability distribution for the number of top to bottom spanning clusters in Directed percolation in two and three dimensions appears to be universal and is of the form $P(n) \sim \exp(-\alpha n^2)$. We argue that $\alpha$ is a new…
We consider fluctuations in the distribution of critical points - saddle points, minima and maxima - of random gaussian fields. We calculate the asymptotic limits of the two point correlation function for various critical point densities,…
A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold $p_c\approx 0.655$ is found between…
Second-order measures, such as the two-point correlation function, are geometrical quantities describing the clustering properties of a point distribution. In this article well-known estimators for the correlation integral are reviewed and…
We show by numerical simulations that the correlation function of the random field Ising model (RFIM) in the critical region in three dimensions has very strong fluctuations and that in a finite volume the correlation length is not…
The Fortuin-Kasteleyn mapping between the Ising model and the site-bond correlated percolation model is shown to be only one of an infinite class of exact mappings. These new cluster representations are a result of "renormalized"…
We consider the Ising model on the square lattice with biaxially correlated random ferromagnetic couplings, the critical point of which is fixed by self-duality. The disorder represents a relevant perturbation according to the extended…
In this paper we establish some relations between percolation on a given graph G and its geometry. Our main result shows that, if G has polynomial growth and satisfies what we call the local isoperimetric inequality of dimension d > 1, then…
We study the dynamical properties of the random transverse-field Ising chain at criticality using a mapping to free fermions, with which we can obtain numerically exact results for system sizes, L, as large as 256. The probability…
The correlation function of two dimensional Ising model with the nearest neighbours interaction on the finite size lattice with the periodical boundary conditions is derived. The expressions similar to the form factor representation are…
Numerical investigation of critical exponents on a hypercubic with L^d random sites with L up to $33 and d up to 7 show that above the critical dimension the phase transitions in Ising model and percolation are not alike.