Black Holes and Random Variables

What the math gives to ML

The paper identifies a universal extreme-value law for maxima of centered log-correlated spectral fields: the maximum grows as a leading log N term with a sharp negative three-quarter log log N correction, followed by an order-one fluctuation. The transferable asset is the explicit finite-size calibration and the heavy extreme tail P(Y greater than y) approximately y exp(-2y). A practical neural-network adaptation is to treat attention-logit or activation fields as approximately correlated random fields and regularize their maxima against this calibrated baseline, avoiding crude global clipping. This can test whether controlling rare logit spikes improves stability and calibration while preserving useful attention sparsity.