Sharp threshold sequence and universality for Ising perceptron models

We study a family of Ising perceptron models with 0,1-valued activation functions. This includes the classical half-space models, as well as some of the symmetric models considered in recent works. For each of these models we show that the free energy is self-averaging, there is a sharp threshold sequence, and the free energy is universal with respect to the disorder. A prior work of C. Xu (2019) used very different methods to show a sharp threshold sequence in the half-space Ising perceptron with Bernoulli disorder. Recent works of Perkins-Xu (2021) and Abbe-Li-Sly (2021) determined the sharp threshold and limiting free energy in a symmetric perceptron model. The results of this paper apply in more general settings, and are based on new “add one constraint” estimates extending Talagrand's estimates for the half-space model (1999, 2011). * The full version of the paper can be accessed at https://arxiv.org/abs/2204.03469.