Sharp Thresholds for Distance Patterns in Random Subsets of ${\mathbb{Z}}^d$

Christina Giannitsi, Eyvindur Ari Palsson

Let $d\ge 5$, $0<\gamma <d-2$, and ${\Omega }_N$ be the binomial random subset of $Q_N=[-N,N{]}^d\cap {\mathbb{Z}}^d$ with retention probability $p_N=N^{-\gamma }$. We prove that, with failure…