Alexander E. Holroyd — Page 5

arXiv2002

Trees and matchings from point processes

A factor graph of a point process is a graph whose vertices are the points of the process, and which is constructed from the process in a deterministic isometry-invariant way. We prove that the d-dimensional Poisson process has a one-ended tree as a factor graph. This implies that the Poisson points can be given an ordering isomorphic to the usual ordering of the integers in a deterministic isometry-invariant way. For d \geq 4 our result answers a question posed by Ferrari, Landim and Thorisson. We prove also that any isometry-invariant ergodic point process of finite intensity in Euclidean or hyperbolic space has a perfect matching as a factor graph provided all the inter-point distances are distinct.

math.PR↗

arXiv2002

Sharp Metastability Threshold for Two-Dimensional Bootstrap Percolation

In the bootstrap percolation model, sites in an $L$ by $L$ square are initially independently declared active with probability $p$. At each time step, an inactive site becomes active if at least two of its four neighbours are active. We study the behaviour as $p \to 0$ and $L \to \infty$ simultaneously of the probability $I(L,p)$ that the entire square is eventually active. We prove that $I(L,p) \to 1$ if $\liminf p \log L > λ$, and $I(L,p) \to 0$ if $\limsup p \log L < λ$, where $λ= π^2/18$. We prove the same behaviour, with the same threshold $λ$, for the probability $J(L,p)$ that a site is active by time $L$ in the process on the infinite lattice. The same results hold for the so-called modified bootstrap percolation model, but with threshold $λ' = π^2/6$. The existence of the thresholds $λ,λ'$ settles a conjecture of Aizenman and Lebowitz, while the determination of their values corrects numerical predictions of Adler, Stauffer and Aharony.

math.PR↗