Sharp threshold for the Ising perceptron model

Consider the discrete cube $\{-1,1{\}}^N$ and a random collection of half spaces which includes each half space $H(x) := {y \in {-1,1}^N: x \cdot y \geq \kappa \sqrt{N}}$for$x \in {-1,1}^N$independentlywithprobability$p$. Is the intersection of these half spaces empty? This is called the Ising perceptron model under Bernoulli disorder. We prove that this event has a sharp threshold; that is, the probability that the intersection is empty increases quickly from $ϵ$ to $1-ϵ$ when $p$ increases only by a factor of $1+o(1)$ as $N\to \mathrm{\infty }$.