Youngtak Sohn

math.PR — Probability

Parisi formula for Ising and Spherical Perceptrons

Contributed by Daniel Fu, Youngtak Sohn

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 Gaussian projections, resolving a conjecture of Bolthausen and Kistler (2010). We also resolve the sharp-threshold conjecture of Aubin, Perkins, and Zdeborová (2019) for the $u$-function binary perceptron at every positive width, and establish the predicted variational formulas for the entropy and capacity of the negative spherical perceptron. Finally, the zero-temperature limit yields the scalar projection-pursuit formulas conjectured by Montanari and Zhou (2025, 2026), in both the supervised and unlabelled settings.