Related papers: Restricted percolation critical exponents in high …
It is known that the critical probability for the percolation transition is not a sharp threshold, actually it is a region of non-zero width $\Delta p_c$ for systems of finite size. Here we present evidence that for complex networks $\Delta…
The finite-size scaling theory for continuous phase transition plays an important role in determining critical point and critical exponents from the size-dependent behaviors of quantities in the thermodynamic limit. For percolation phase…
We consider high-dimensional percolation at the critical threshold. We condition the origin to be disjointly connected to two points, $x$ and $x'$, and subsequently take the limit as $|x|$, $|x'|$ as well as $|x-x'|$ diverge to infinity.…
In this article, we generalize known formulas for crossing probabilities. Prior crossing results date back to J. Cardy's prediction of a formula for the probability that a percolation cluster in two dimensions connects the left and right…
This paper explores the possibility of establishing an analytic form of the distribution of the order parameter fluctuations in a two-dimensional critical spin wave model, or width fluctuations of a two dimensional Edwards-Wilkinson…
In the past years, many properties of the largest connected components of critical percolation on the high-dimensional torus, such as their sizes and diameter, have been established. The order of magnitude of these quantities equals the one…
The methods of conformal field theory are used to compute the crossing probabilities between segments of the boundary of a compact two-dimensional region at the percolation threshold. These probabilities are shown to be invariant not only…
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…
We consider percolation of the vacant set of random interlacements at intensity $u$ in dimensions three and higher, and derive lower bounds on the truncated two-point function for all values of $u>0$. These bounds are sharp up to principal…
We consider long-range percolation, Ising model, and self-avoiding walk on $\mathbb{Z}^d$, with couplings decaying like $|x|^{-(d+\alpha)}$ where $0 < \alpha \le 2$, above the upper critical dimensions. In the spread-out setting where the…
We consider random interlacements on $ \mathbb{Z}^d$, $d \ge 3$, when their vacant set is in a strongly percolative regime. Given a large box centered at the origin, we establish an asymptotic upper bound on the exponential rate of decay of…
We prove that for the Gaussian free field (GFF) on the metric graph of $\mathbb{Z}^d$ (for all $d\ge 3$ except the critical dimension $d_c=6$), with uniformly positive probability there exist two distinct sign clusters of diameter at least…
Using an ensemble of high resolution 2D numerical simulations, we explore the scaling properties of cosmological density fluctuations in the non-linear regime. We study the scaling behaviour of the usual $N$--point volume-averaged…
There are various notions of dimension in fractal geometry to characterise (random and non-random) subsets of $\mathbb R^d$. In this expository text, we discuss their analogues for infinite subsets of $\mathbb Z^d$ and, more generally, for…
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…
We present a study of connectivity percolation in suspensions of hard spherocylinders by means of Monte Carlo simulation and connectedness percolation theory. We focus attention on polydispersity in the length, the diameter and the…
We study the center structure of full dynamical QCD at finite temperatures and nonzero values of the background magnetic field using continuum extrapolated lattice data. We concentrate on two particular observables characterizing center…
For ordinary (independent) percolation on a large class of lattices it is well known that below the critical percolation parameter $p_c$ the cluster size distribution has exponential decay and that power-law behavior of this distribution…
We consider long-range self-avoiding walk, percolation and the Ising model on $\mathbb{Z}^d$ that are defined by power-law decaying pair potentials of the form $D(x)\asymp|x|^{-d-\alpha}$ with $\alpha>0$. The upper-critical dimension…
A fundamental property of complex networks is the tendency for edges to cluster. The extent of the clustering is typically quantified by the clustering coefficient, which is the probability that a length-2 path is closed, i.e., induces a…