Exact storage capacity of dense associative memories with p-body interactions
In plain words
A Hopfield network stores patterns as low points of an energy, and replacing pairwise couplings by couplings among p units at once lets it store vastly more patterns. The exact number of patterns it can store and still recall from a corrupted cue has not been computed with proof.
Precise statement
Spins $\sigma_i = \pm 1, i = 1..N$, energy $E = -N \sum_{\mu=1..P} (m_\mu)^p$ with overlaps $m_\mu = (1/N) \sum_i \xi_i^\mu \sigma_i$ and P independent uniform random patterns $\xi^\mu$, at load $P = \alpha N^{(p-1)}$ with $p \ge 3$ fixed. Determine $\alpha_c(p)$ such that for $\alpha < \alpha_c(p)$ every pattern has a nearby energy minimum with overlap m close to 1 and a basin of attraction of macroscopic size under zero-temperature single-spin-flip dynamics, and for $\alpha > \alpha_c(p)$ it does not. The answer is $\alpha_c(p)$ with a rigorous proof, or a proof that the replica-method prediction is exact.
What would settle it
A rigorous computation of $\alpha_{c}(p)$ for the retrieval transition at load $P$ proportional to $N^{p-1}$, with basins of macroscopic size.
Status in the literature
Unverified note
The scaling $P \sim N^{p-1}$ and a replica prediction of $\alpha_{c}(p)$ date to 1987 (Gardner, J. Phys. A 20, 3453, DOI 10.1088/0305-4470/20/11/046), the model was revived as dense associative memory by Krotov and Hopfield (2016), fixed-point capacity of order $N^{p-1}/\ln N$ is proved, and the exponential-interaction version was solved by replica methods (Lucibello and Mezard, PRL 2024); no rigorous derivation of $\alpha_{c}(p)$ at P proportional to $N^{p-1}$ was located (2026).