Sharp threshold for the Ising perceptron model

Abstract

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

19 pages

Cited by in corpus (1)

• Frozen $1$-RSB structure of the symmetric Ising perceptron