Related papers: Critical speeding-up in dynamical percolation
Consider a cellular automaton with state space $\{0,1 \}^{{\mathbb Z}^2}$ where the initial configuration $\omega_0$ is chosen according to a Bernoulli product measure, 1's are stable, and 0's become 1's if they are surrounded by at least…
The divergence of the correlation length $\xi$ at criticality is an important phenomenon of percolation in two-dimensional systems. Substantial speed-ups to the calculation of the percolation threshold and component distribution have been…
We consider biased random walks in a one-dimensional percolation model. This model goes back to Axelson-Fisk and H\"aggstr\"om and exhibits the same phase transition as biased random walk on the infinite cluster of supercritical Bernoulli…
We consider the model of a directed polymer in a random environment defined on the infinite cluster of supercritical Bernoulli bond percolation in dimensions $d \geq 3$. For this model, it was proved in arXiv:2205.06206 that for almost…
We re-examine a two-dimensional forest-fire model via Monte-Carlo simulations and show the existence of two length scales with different critical exponents associated with clusters and with the usual two-point correlation function of trees.…
A region of two-dimensional space has been filled randomly with large number of growing circular discs allowing only a `slight' overlapping among them just before their growth stop. More specifically, each disc grows from a nucleation…
This is the first of two papers on the critical behaviour of bond percolation models in high dimensions. In this paper, we obtain strong joint control of the critical exponents eta and delta, for the nearest-neighbour model in very high…
Recently, the diffusion-limited cluster aggregation (DLCA) model was restudied as a real-world example of showing discontinuous percolation transitions (PTs). Because a larger cluster is less mobile in Brownian motion, it comes into contact…
We investigate the problem of growing clusters, which is modeled by two dimensional disks and three dimensional droplets. In this model we place a number of seeds on random locations on a lattice with an initial occupation probability, $p$.…
Transient homogeneous nucleation is studied in the limit of large critical sizes. Starting from pure monomers, three eras of transient nucleation are characterized in the classic Becker-D\"oring kinetic equations with the Turnbull-Fisher…
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.…
The close analogy between cluster percolation and string proliferation in the context of critical phenomena is studied. Like clusters in percolation theory, closed strings, which can be either finite-temperature worldlines or topological…
We consider the Boolean model $Z$ on $\mathbb{R}^d$ with random compact grains, i.e. $Z := \bigcup_{i \in \mathbb{N}} (X_i + Z_i)$ where $\eta_t := \{X_1, X_2, \dots\}$ is a Poisson point process of intensity $t$ and $(Z_1, Z_2, \dots)$ is…
This paper studies growth, percolation, and correlations in disordered fiber networks. We start by introducing a 2D continuum deposition model with effective fiber-fiber interactions represented by a parameter $p$ which controls the degree…
In this note we provide an alternative proof of the fact that subcritical bootstrap percolation models have a positive critical probability in any dimension. The proof relies on a recent extension of the classical framework of Toom. This…
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…
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…
We study the response of a nematic colloidal dispersion of rods to a driven probe particle which is dragged with high speed through the dispersion perpendicular to the nematic director. In front of the dragged particle, clusters of rods are…
Let C_n be the origin-containing cluster in subcritical percolation on the lattice (1/n) Z^d, viewed as a random variable in the space Omega of compact, connected, origin-containing subsets of R^d, endowed with the Hausdorff metric delta.…
In this manuscript, we address open questions raised by Dieker \& Yakir (2014), who proposed a novel method of estimation of (discrete) Pickands constants $\mathcal{H}^\delta_\alpha$ using a family of estimators $\xi^\delta_\alpha(T), T>0$,…