Sharp threshold for the Ising perceptron model

Abstract

Consider the discrete cube ${\{-1,1\}}^{\mathit{N}}$ and a random collection of half spaces which includes each half space $\mathit{H}(\mathit{x}):=\{\mathit{y}\in {\{-1,1\}}^{\mathit{N}}:\mathit{x}·\mathit{y}\ge \mathit{\kappa }\sqrt{\mathit{N}}\}$ for $\mathit{x}\in {\{-1,1\}}^{\mathit{N}}$ independently with probability 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 $1-\mathit{ϵ}$ when p increases only by a factor of $1+\mathit{o}(1)$ as $\mathit{N}\to \infty$.