Free energy fluctuations of symmetric perceptron models up to and including criticality
Abstract
We study the free energy fluctuations of spherical and Ising perceptron models with general even activations. These include the symmetric binary perceptron, the U-perceptron, and the bipartite Sherrington--Kirkpatrick model. Our main result is a Gaussian CLT for the free energy that holds both in the ``high-temperature'' interior of a second moment RS regime and on its ``critical'' boundary. At criticality, the CLT has scale ((log N)/6)1/2, which matches the free energy fluctuations of the critical Sherrington--Kirkpatrick model and suggests this scaling is universal. The proof shows that the perceptron model's free energy fluctuations are explained, up to an additive o(1) error with high probability, by those of a corresponding spherical Hopfield model, and that this relation persists even at criticality. This proof strategy builds on a critical reweighted moment method developed by the first and third authors \cite{du2026fluctuations} to identify the critical fluctuations of the SK free energy, which links these fluctuations to the spectral edge spacing of the disorder.
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BibTeX
@misc{du2026free,
title = {{Free energy fluctuations of symmetric perceptron models up to and including criticality}},
author = {Hang Du and Shuyang Gong and Brice Huang and Zhangsong Li},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.31736},
eprint = {2610.31736},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.31736},
}
RIS
TY - UNPB
TI - Free energy fluctuations of symmetric perceptron models up to and including criticality
AU - Hang Du
AU - Shuyang Gong
AU - Brice Huang
AU - Zhangsong Li
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.31736
UR - https://arxiv.org/abs/2610.31736
AB - We study the free energy fluctuations of spherical and Ising perceptron models with general even activations. These include the symmetric binary perceptron, the U-perceptron, and the bipartite Sherrington--Kirkpatrick model. Our main result is a Gaussian CLT for the free energy that holds both in the ``high-temperature'' interior of a second moment RS regime and on its ``critical'' boundary. At criticality, the CLT has scale ((log N)/6)^1/2, which matches the free energy fluctuations of the critical Sherrington--Kirkpatrick model and suggests this scaling is universal. The proof shows that the perceptron model's free energy fluctuations are explained, up to an additive o(1) error with high probability, by those of a corresponding spherical Hopfield model, and that this relation persists even at criticality. This proof strategy builds on a critical reweighted moment method developed by the first and third authors \cite{du2026fluctuations} to identify the critical fluctuations of the SK free energy, which links these fluctuations to the spectral edge spacing of the disorder.
ER -