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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.
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We study the independent alignment percolation model on $mathbb{Z}^d$ introduced by Beaton, Grimmett and Holmes [arXiv:1908.07203]. It is a model for random intersecting line segments defined as follows. First the sites of $mathbb{Z}^d$ are independently declared occupied with probability $p$ and vacant otherwise. Conditional on the configuration of occupied vertices, consider the set of all line segments that are parallel to the coordinate axis, whose extremes are occupied vertices and that do not traverse any other occupied vertex. Declare independently the segments on this set open with probability $lambda$ and closed otherwise. All the edges that lie on open segments are also declared open giving rise to a bond percolation model in $mathbb{Z}^d$. We show that for any $d geq 2$ and $p in (0,1]$ the critical value for $lambda$ satisfies $lambda_c(p)<1$ completing the proof that the phase transition is non-trivial over the whole interval $(0,1]$. We also show that the critical curve $p mapsto lambda_c(p)$ is continuous at $p=1$, answering a question posed by the authors in [arXiv:1908.07203].
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.
We prove that the probability of crossing a large square in quenched Voronoi percolation converges to 1/2 at criticality, confirming a conjecture of Benjamini, Kalai and Schramm from 1999. The main new tools are a quenched version of the box-crossing property for Voronoi percolation at criticality, and an Efron-Stein type bound on the variance of the probability of the crossing event in terms of the sum of the squares of the influences. As a corollary of the proof, we moreover obtain that the quenched crossing event at criticality is almost surely noise sensitive.
We study a dependent site percolation model on the $n$-dimensional Euclidean lattice where, instead of single sites, entire hyperplanes are removed independently at random. We extend the results about Bernoulli line percolation showing that the model undergoes a non-trivial phase transition and proving the existence of a transition from exponential to power-law decay within some regions of the subcritical phase.
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