Parisi formula for Ising and Spherical Perceptrons
We prove a Parisi formula for Ising and spherical perceptrons with Gaussian disorder and i.i.d. random potentials having a common Lipschitz bound and integrable values at zero. We assume $M/N\to\alpha\in(0,\infty)$, where $M$ is the number of Gaussian vectors and $N$ is the number of spins. The proof combines cavity in $M$ and $N$ with the Aizenman–Sims–Starr scheme for the lower bound and Guerra’s interpolation for the upper bound. As consequences, we answer several problems posed by Talagrand (2011) and establish a quenched large deviation principle for the empirical distribution of the…
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