Krauth–Mézard conjecture on the capacity of the binary perceptron

Establish that for a random matrix A ∈ ℝ^{αN×N} with i.i.d. standard Gaussian entries, the capacity α_*—defined as the largest α for which there exists σ ∈ {−1,+1}^N such that Aσ > 0 entrywise—concentrates around an explicit constant α_c ≈ 0.833 as N → ∞.

Background

The binary perceptron asks whether, for a given α, there exists a binary vector σ ∈ {−1,+1}^N satisfying Aσ > 0 entrywise, where A has i.i.d. standard Gaussian entries. The capacity α* is the largest such α, and the long-standing problem is to determine the typical value α_c around which α* concentrates.

Krauth and Mézard (1989) conjectured, via physics-inspired TAP (Thouless–Anderson–Palmer) heuristics, that α_c ≈ 0.833. Rigorous progress includes a matching lower bound (conditional on numerical assumptions) by Ding and Sun (2019) and upper bounds of 0.9963 by Kim and Roche (1998) and a non-explicit improvement by Talagrand (1999). This note records a complete proof of a sharper upper bound α_c < 0.847, but the exact value remains conjectural.