Abstract
Consider the discrete cube and a random collection of half spaces which includes each half space for 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 when p increases only by a factor of as .
Acknowledgments
The first draft of the paper was completed while the author was a graduate student in the Department of Statistics at the University of Chicago. The author would like to thank Jian Ding for suggesting the problem and carefully reviewing the draft of the paper. He would also like to thank the Associate Editor and the anonymous referee for their careful reading of the paper and numerous valuable suggestions.
Citation
Changji Xu. "Sharp threshold for the Ising perceptron model." Ann. Probab. 49 (5) 2399 - 2415, September 2021. https://doi.org/10.1214/21-AOP1511
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