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We consider two-dimensional dependent dynamical site percolation where sites perform majority dynamics. We introduce the critical percolation function at time t as the infimum density with which one needs to begin in order to obtain an infinite open component at time t. We prove that, for any fixed time t, there is no percolation at criticality and that the critical percolation function is continuous. We also prove that, for any positive time, the percolation threshold is strictly smaller than the critical probability for independent site percolation.
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We study an interacting particle system in which moving particles activate dormant particles linked by the components of critical bond percolation. Addressing a conjecture from Beckman, Dinan, Durrett, Huo, and Junge for a continuous variant, we prove that the process can reach infinity in finite time i.e., explode. In particular, we prove that explosions occur almost surely on regular trees as well as oriented and unoriented two-dimensional integer lattices with sufficiently many particles per site. The oriented case requires an additional hypothesis about the existence and value of a certain critical exponent. We further prove that the process with one particle per site expands at a superlinear rate on integer lattices of any dimension. Some arguments use connections to critical first passage percolation, including a new result about the existence of an infinite path with finite passage time on the oriented two-dimensional lattice.
We consider the median dynamics process in general graphs. In this model, each vertex has an independent initial opinion uniformly distributed in the interval [0,1] and, with rate one, updates its opinion to coincide with the median of its neighbors. This process provides a continuous analog of majority dynamics. We deduce properties of median dynamics through this connection and raise new conjectures regarding the behavior of majority dynamics on general graphs. We also prove these conjectures on some graphs where majority dynamics has a simple description.
The existence (or not) of infinite clusters is explored for two stochastic models of intersecting line segments in $d ge 2$ dimensions. Salient features of the phase diagram are established in each case. The models are based on site percolation on ${mathbb Z}^d$ with parameter $pin (0,1]$. For each occupied site $v$, and for each of the $2d$ possible coordinate directions, declare the entire line segment from $v$ to the next occupied site in the given direction to be either blue or not blue according to a given stochastic rule. In the one-choice model, each occupied site declares one of its $2d$ incident segments to be blue. In the independent model, the states of different line segments are independent.
We define a continuum percolation model that provides a collection of random ellipses on the plane and study the behavior of the covered set and the vacant set, the one obtained by removing all ellipses. Our model generalizes a construction that appears implicitly in the Poisson cylinder model of Tykesson and Windisch. The ellipses model has a parameter $alpha > 0$ associated with the tail decay of the major axis distribution; we only consider distributions $rho$ satisfying $rho[r, infty) asymp r^{-alpha}$. We prove that this model presents a double phase transition in $alpha$. For $alpha in (0,1]$ the plane is completely covered by the ellipses, almost surely. For $alpha in (1,2)$ the vacant set is not empty but does not percolate for any positive density of ellipses, while the covered set always percolates. For $alpha in (2, infty)$ the vacant set percolates for small densities of ellipses and the covered set percolates for large densities. Moreover, we prove for the critical parameter $alpha = 2$ that there is a non-degenerate interval of density for which the probability of crossing boxes of a fixed proportion is bounded away from zero and one, a rather unusual phenomenon. In this interval neither the covered set nor the vacant set percolate, a behavior that is similar to critical independent percolation on $mathbb{Z}^2$.
We study bond percolation on the square lattice with one-dimensional inhomogeneities. Inhomogeneities are introduced in the following way: A vertical column on the square lattice is the set of vertical edges that project to the same vertex on $mathbb{Z}$. Select vertical columns at random independently with a given positive probability. Keep (respectively remove) vertical edges in the selected columns, with probability $p$, (respectively $1-p$). All horizontal edges and vertical edges lying in unselected columns are kept (respectively removed) with probability $q$, (respectively $1-q$). We show that, if $p > p_c(mathbb{Z}^2)$ (the critical point for homogeneous Bernoulli bond percolation) then $q$ can be taken strictly smaller then $p_c(mathbb{Z}^2)$ in such a way that the probability that the origin percolates is still positive.
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