Related papers: Sharp Phase Transition for the Formation of Infini…

In an investigation of percolation on isoradial graphs, we prove the criticality of canonical bond percolation on isoradial embeddings of planar graphs, thus extending celebrated earlier results for homogeneous and inhomogeneous square,…

Shape-dependent universal crossing probabilities are studied, via Monte Carlo simulations, for bond and site directed percolation on the square lattice in the diagonal direction, at the percolation threshold. Since the system is strongly…

In a previous paper, we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph, and showed that this triangle condition implies that the…

Let $(G_n) = \left((V_n,E_n)\right)$ be a sequence of finite connected vertex-transitive graphs with uniformly bounded vertex degrees such that $\lvert V_n \rvert \to \infty$ as $n \to \infty$. We say that percolation on $G_n$ has a sharp…

The orthant model is a directed percolation model on $\mathbb{Z}^d$, in which all clusters are infinite. We prove a sharp threshold result for this model: if $p$ is larger than the critical value above which the cluster of $0$ is contained…

We introduce a bond percolation procedure on a $D$-dimensional lattice where two neighbouring sites are connected by $N$ channels, each operated by valves at both ends. Out of a total of $N$, randomly chosen $n$ valves are open at every…

Percolation problems appear in a large variety of different contexts ranging from the design of composite materials to vaccination strategies on community networks. The key observable for many applications is the percolation threshold.…

Explosive percolation (EP) has received significant research attention due to its rich and anomalous phenomena near criticality. In our recent study [Phys. Rev. Lett. 130, 147101 (2023)], we demonstrated that the correct critical behaviors…

Bootstrap percolation on a graph iteratively enlarges a set of occupied sites by adjoining points with at least $\theta$ occupied neighbors. The initially occupied set is random, given by a uniform product measure, and we say that spanning…

We study the history-dependent percolation in two dimensions, which evolves in generations from standard bond-percolation configurations through iteratively removing occupied bonds. Extensive simulations are performed for various…

We consider supercritical bond percolation in $\mathbb{Z}^d$ for $d \geq 3$. The origin lies in a finite open cluster with positive probability, and, when it does, the diameter of this cluster has an exponentially decaying tail. For each…

We study the probability that a loop is null-homotopic -- that is, bounded by the continuous image of a disk -- in plaquette percolation on $\mathbb{Z}^3.$ Locally, the event that there is a ``horizontal disk crossing'' of a rectangular…

We describe the critical window for percolation in the universality class of sparse growing random graphs. In our models, vertices arrive sequentially and connect independently to each earlier vertex $v$ with probability proportional to a…

In the Constrained-degree percolation model on a graph $(\mathbb{V},\mathbb{E})$ there are a sequence, $(U_e)_{e\in\mathbb{E}}$, of i.i.d. random variables with distribution $U[0,1]$ and a positive integer $k$. Each bond $e$ tries to open…

Let $(G_n)$ be a sequence of finite connected vertex-transitive graphs with volume tending to infinity. We say that a sequence of parameters $(p_n)$ is a percolation threshold if for every $\varepsilon > 0$, the proportion $\left\lVert K_1…

We study percolation properties of the upper invariant measure of the contact process on $\mathbb{Z}^d$. Our main result is a sharp percolation phase transition with exponentially small clusters throughout the subcritical regime and a…

Percolation is perhaps the simplest example of a process exhibiting a phase transition and one of the most studied phenomena in statistical physics. The percolation transition is continuous if sites/bonds are occupied independently with the…

There have been several spectral bounds for the percolation transition in networks, using spectrum of matrices associated with the network such as the adjacency matrix and the non-backtracking matrix. However they are far from being tight…

Recently, the number of non-standard percolation models has proliferated. In all these models, there exists a phase transition at which long range connectivity is established, if local connectedness increases through a threshold $p_c$. In…

Let a random geometric graph be defined in the supercritical regime for the existence of a unique infinite connected component in Euclidean space. Consider the first-passage percolation model with independent and identically distributed…