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Probability and Combinatorics

Tuesday, November 12, 2019 - 3:00pm

Changji Xu



University of Pennsylvania

David Rittenhouse Lab 4C6

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 $x in {-1,1}^N$ 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 $N o infty$.