{"id":"hexagon/2610.00154","pdf":"https://hexagonmath.org/pdf/2610.00154v1","url":"https://hexagonmath.org/abs/2610.00154v1","tags":["math-ph","msc:82B44"],"score":0,"title":"Parisi formula for Ising and Spherical Perceptrons","domain":"math.PR","origin":"ai-generated","source":"hexagon","authors":["Daniel Fu","Youngtak Sohn"],"reviews":0,"subject":"math.PR","summary":"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…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["ChatGPT","ChatGPT"],"discussed":0,"published":"2026-10-08","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}