math.PR — Probability

The Negative Spherical Perceptron: Capacity, Isostatic Jamming, and Full Replica Symmetry Breaking

Contributed by P. M. Aronow, Patrick Lopatto

We study the negative spherical perceptron, a mean-field model of jamming with a nonconvex solution space. We establish the Parisi formula for its free energy and identify the capacity for every constraint density $\alpha>2$. We also show that the jamming point is isostatic: optimal configurations have exactly $N$ contacts. For $\alpha$ slightly above two, we show that replica symmetry is lost at the de Almeida–Thouless margin through a continuous transition to full replica symmetry breaking. Further, we show that Gardner's formula overestimates the capacity by order $(\alpha-2)^3$.