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Register a new userSite Percolation on a Disordered Triangulation of the Square Lattice
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Leonardo T. Rolla
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In this paper we consider independent site percolation in a triangulation of $mathbb{R}^2$ given by adding $sqrt{2}$-long diagonals to the usual graph $mathbb{Z}^2$. We conjecture that $p_c=frac{1}{2}$ for any such graph, and prove it for almost every such graph.
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On the square lattice raindrops fall on an edge with midpoint $x$ at rate $|x|_infty^{-alpha}$. The edge becomes open when the first drop falls on it. Let $rho(x,t)$ be the probability that the edge with midpoint $x=(x_1,x_2)$ is open at time $t$ and let $n(p,t)$ be the distance at which edges are open with probability $p$ at time $t$. We show that with probability tending to 1 as $t to infty$: (i) the cluster containing the origin $mathbb C_0(t)$ is contained in the square of radius $n(p_c-epsilon,t)$, and (ii) the cluster fills the square of radius $n(p_c+epsilon,t)$ with the density of points near $x$ being close to $theta(rho(x,t))$ where $theta(p)$ is the percolation probability when bonds are open with probability $p$ on $mathbb Z^2$. Results of Nolin suggest that if $N=n(p_c,t)$ then the boundary fluctuations of $mathbb C_0(t)$ are of size $N^{4/7}$.
In Poisson percolation each edge becomes open after an independent exponentially distributed time with rate that decreases in the distance from the origin. As a sequel to our work on the square lattice, we describe the limiting shape of the component containing the origin in the oriented case. We show that the density of occupied sites at height $y$ in the cluster is close to the percolation probability in the corresponding homogeneous percolation process, and we study the fluctuations of the boundary.
Bootstrap percolation on a graph is a deterministic process that 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 with a low density $p$. Our main focus is on this process on the product graph $mathbb{Z}^2times K_n^2$, where $K_n$ is a complete graph. We investigate how $p$ scales with $n$ so that a typical site is eventually occupied. Under critical scaling, the dynamics with even $theta$ exhibits a sharp phase transition, while odd $theta$ yields a gradual percolation transition. We also establish a gradual transition for bootstrap percolation on $mathbb{Z}^2times K_n$. The main tool is heterogeneous bootstrap percolation on $mathbb{Z}^2$.
We prove that for a non-amenable, locally finite, connected, transitive, planar graph with one end, any automorphism invariant site percolation on the graph does not have exactly 1 infinite 1-cluster and exactly 1 infinite 0-cluster a.s. If we further assume that the site percolation is insertion-tolerant and a.s.~there exists a unique infinite 0-cluster, then a.s.~there are no infinite 1-clusters. The proof is based on the analysis of a class of delicately constructed interfaces between clusters and contours. Applied to the case of i.i.d.~Bernoulli site percolation on infinite, connected, locally finite, transitive, planar graphs, these results solve two conjectures of Benjamini and Schramm (Conjectures 7 and 8 in cite{bs96}) in 1996.
Chase-escape percolation is a variation of the standard epidemic spread models. In this model, each site can be in one of three states: unoccupied, occupied by a single prey, or occupied by a single predator. Prey particles spread to neighboring empty sites at rate $p$, and predator particles spread only to neighboring sites occupied by prey particles at rate $1$, killing the prey particle that existed at that site. It was found that the prey can survive with non-zero probability, if $p>p_c$ with $p_c<1$. Using Monte Carlo simulations on the square lattice, we estimate the value of $p_c = 0.49451 pm 0.00001$, and the critical exponents are consistent with the undirected percolation universality class. We define a discrete-time parallel-update version of the model, which brings out the relation between chase-escape and undirected bond percolation. For all $p < p_c$ in $D$-dimensions, the number of predators in the absorbing configuration has a stretched-exponential distribution in contrast to the exponential distribution in the standard percolation theory. We also study the problem starting from the line initial condition with predator particles on all lattice points of the line $y=0$ and prey particles on the line $y=1$. In this case, for $p_c<p < 1$, the center of mass of the fluctuating prey and predator fronts travel at the same speed. This speed is strictly smaller than the speed of an Eden front with the same value of $p$, but with no predators. At $p=1$, the fronts undergo a depinning transition. The fluctuations of the front follow Kardar-Parisi-Zhang scaling both above and below this depinning transition.
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