Free energy fluctuations of symmetric perceptron models up to and including criticality

arXiv:2610.31736v1 Announce Type: new 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 $(\frac{\log N}{6})^{1/2}$, 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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