Related papers: Varying critical percolation exponents on a multif…
We give a self-contained and detailed presentation of Kesten's results that allow to relate critical and near-critical percolation on the triangular lattice. They constitute an important step in the derivation of the exponents describing…
We numerically investigate the heterogeneity in cluster sizes in the two-dimensional Ising model and verify its scaling form recently proposed in the context of percolation problems [Phys. Rev. E 84, 010101(R) (2011)]. The scaling exponents…
The properties of the similarity transformation in percolation theory in the complex plane of the percolation probability are studied. It is shown that the percolation problem on a two-dimensional square lattice reduces to the Mandelbrot…
A multifractal model is used to analyze neutron evolution within a reactor. For chain reactions, various characteristics of multifractal neutron behavior have been determined. These include the dimension of the multifractal carrier,…
We investigate the component sizes of the critical configuration model, as well as the related problem of critical percolation on a supercritical configuration model. We show that, at criticality, the finite third moment assumption on the…
The following article deals with the critical value p_c of the three-dimensional bootstrap percolation. We will check the behavior of p_c for different lengths of the lattice and additionally we will scale p_c in the limit of an infinite…
The concept of midpoint percolation has recently been applied to characterize the double percolation transitions in negatively curved structures. Regular $d$-dimensional hypercubic lattices are in the present work investigated using the…
The fractal structure and scaling properties of a 2d slice of the 3d Ising model is studied using Monte Carlo techniques. The percolation transition of geometric spin (GS) clusters is found to occur at the Curie point, reflecting the…
We study a percolation model on the square lattice, where clusters "freeze" (stop growing) as soon as their volume (i.e. the number of sites they contain) gets larger than N, the parameter of the model. A model where clusters freeze when…
It is believed that the large-scale geometric properties of two-dimensional critical percolation are described by a logarithmic conformal field theory, but it has been challenging to exhibit concrete examples of logarithmic singularities…
The rigidity transition occurs when, as the density of microscopic components is increased, a disordered medium becomes able to transmit and ensure macroscopic mechanical stability, owing to the appearance of a space-spanning rigid…
For percolating systems, we propose a universal exponent relation connecting the leading corrections to scaling of the cluster size distribution with the dynamic corrections to the asymptotic transport behaviour at criticality. Our…
It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of…
We show that the correction-to-scaling exponents in two-dimensional percolation are bounded by Omega <= 72/91, omega = D Omega <= 3/2, and Delta_1 = nu omega <= 2, based upon Cardy's result for the critical crossing probability on an…
The critical behavior at a corner in two-dimensional Ising and three-state Potts models is studied numerically on the square lattice using transfer operator techniques. The local critical exponents for the magnetization and the energy…
We study the boundary effects in invasion percolation with and without trapping. We find that the presence of boundaries introduces a new set of surface critical exponents, as in the case of standard percolation. Numerical simulations show…
We present an "ultimate" proof of Cardy's formula for the critical percolation on the hexagonal lattice \cite{Smirnov01criticalpercolation}, showing the existence of the universal and conformally invariant scaling limit of crossing…
We study the fractal structure of Diffusion-Limited Aggregation (DLA) clusters on the square lattice by extensive numerical simulations (with clusters having up to $10^8$ particles). We observe that DLA clusters undergo strongly anisotropic…
In two space dimensions, the percolation point of the pure-site clusters of the Ising model coincides with the critical point T_c of the thermal transition and the percolation exponents belong to a special universality class. By introducing…
We present a review of the recent progress on percolation scaling limits in two dimensions. In particular, we will consider the convergence of critical crossing probabilities to Cardy's formula and of the critical exploration path to…