Related papers: Sharpness of the Phase Transition for the Orthant …

We establish the sharpness of the percolation phase transition for a class of infinite-range weighted random connection models. The vertex set is given by a marked Poisson point process on $\mathbb{R}^d$ with intensity $\lambda>0$, where…

In the corrupted compass model on a vertex-transitive graph, a neighbouring edge of every vertex is chosen uniformly at random and opened. Additionally, with probability $p$, independently for every vertex, every neighbouring edge is…

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…

We establish that the phase transition for infinite cycles in the random stirring model on an infinite regular tree of high degree is sharp. That is, we prove that there exists d_0 such that, for any d \geq d_0, the set of parameter values…

We study a particular model of a random medium, called the orthant model, in general dimensions $d\ge 2$. Each site $x\in \Z^d$ independently has arrows pointing to its positive neighbours $x+e_i$, $i=1,\dots, d$ with probability $p$ and…

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 prove the sharpness of the phase transition for speed in the biased random walk on the supercritical percolation cluster on Z^d. That is, for each d at least 2, and for any supercritical parameter p > p_c, we prove the existence of a…

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…

We provide a new proof of the sharpness of the phase transition for nearest-neighbour Bernoulli percolation. More precisely, we show that - for $p<p_c$, the probability that the origin is connected by an open path to distance $n$ decays…

The half-orthant model is a partially oriented model of a random medium involving a parameter $p\in [0,1]$, for which there is a critical value $p_c(d)$ (depending on the dimension $d$) below which every point is reachable from the origin.…

We prove an inequality on decision trees on monotonic measures which generalizes the OSSS inequality on product spaces. As an application, we use this inequality to prove a number of new results on lattice spin models and their…

One of the most well-known classical results for site percolation on the square lattice is the equation p_c + p_c^* = 1. In words, this equation means that for all values different from p_c of the parameter p the following holds: Either…

We prove that random-cluster models with q larger than 1 on a variety of planar lattices have a sharp phase transition, that is that there exists some parameter p_c below which the model exhibits exponential decay and above which there…

Classical bond percolation theory studies the conditions for a given point in a random graph to be connected to infinity, or "escape" to infinity, via a sequence of random edges. In this work, we present a higher-dimensional generalization…

In this paper, we consider Bernoulli percolation on a locally finite, transitive and infinite graph (e.g. the hypercubic lattice $\mathbb{Z}^d$). We prove the following estimate, where $\theta_n(p)$ is the probability that there is a path…

One of the most well-known classical results for site percolation on the square lattice is the equation $p_c+p_c^*=1$. In words, this equation means that for all values $\neq p_c$ of the parameter $p$, the following holds: either a.s. there…

In a new type of percolation phase transition, which was observed in a set of non-equilibrium models, each new connection between vertices is chosen from a number of possibilities by an Achlioptas-like algorithm. This causes preferential…

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…

We study versions of the contact process with three states, and with infections occurring at a rate depending on the overall infection density. Motivated by a model described in [17] for vegetation patterns in arid landscapes, we focus on…

We study a model of an i.i.d.~random environment in general dimensions $d\ge 2$, where each site is equipped with one of two environments. The model comes with a parameter $p$ which governs the frequency of the first environment, and for…