We summarize our notations in the following table for easy reference.

We introduce a three-dimensional vectorial extension of the Hopfield associative-memory model in which each neuron is a unit vector on $S^2$ and synaptic couplings are $3\times 3$ blocks generated through a vectorial Hebbian rule. The resulting block-structured operator is mathematically analogous to the Hessian of amorphous solids and induces a rigid energy landscape with deep minima for stored patterns. Simulations and spectral analysis show that the vectorial network substantially outperforms the classical binary Hopfield model. For moderate connectivity, the critical storage ratio $γ_c$ grows approximately linearly with the coordination number $Z$, while for $Z\gtrsim 40$ a high-connectivity regime emerges in which $γ_c$ systematically exceeds the extrapolated low-$Z$ linear fit. At the same time, a persistent spectral gap separates pattern modes from the bulk and basins of attraction enlarge, yielding enhanced robustness to initialization noise. Thus geometric constraints combined with amorphous-solid-inspired structure produce associative memories with superior storage and retrieval performance, especially in the high-connectivity ($Z \gtrsim 20$-$30$) regime.

We investigate the phase diagram and memory retrieval capabilities of bipartite energy-based neural networks, namely Restricted Boltzmann Machines (RBMs), as a function of the prior distribution imposed on their hidden units - including binary, multi-state, and ReLU-like activations. Drawing connections to the Hopfield model and employing analytical tools from statistical physics of disordered systems, we explore how the architectural choices and activation functions shape the thermodynamic properties of these models. Our analysis reveals that standard RBMs with binary hidden nodes and extensive connectivity suffer from reduced critical capacity, limiting their effectiveness as associative memories. To address this, we examine several modifications, such as introducing local biases and adopting richer hidden unit priors. These adjustments restore ordered retrieval phases and markedly improve recall performance, even at finite temperatures. Our theoretical findings, supported by finite-size Monte Carlo simulations, highlight the importance of hidden unit design in enhancing the expressive power of RBMs.

We study the optimal memorization capacity of modern Hopfield models and Kernelized Hopfield Models (KHMs), a transformer-compatible class of Dense Associative Memories.

We solve the mean field equations for a stochastic Hopfield network with temperature (noise) in the presence of strong, i.e., multiply stored, patterns, and use this solution to obtain the storage capacity of such a network. Our result provides for the first time a rigorous solution of the mean filed equations for the standard Hopfield model and is in contrast to the mathematically unjustifiable replica technique that has been used hitherto for this derivation. We show that the critical temperature for stability of a strong pattern is equal to its degree or multiplicity, when the sum of the squares of degrees of the patterns is negligible compared to the network size. In the case of a single strong pattern, when the ratio of the number of all stored pattens and the network size is a positive constant, we obtain the distribution of the overlaps of the patterns with the mean field and deduce that the storage capacity for retrieving a strong pattern exceeds that for retrieving a simple pattern by a multiplicative factor equal to the square of the degree of the strong pattern. This square law property provides justification for using strong patterns to model attachment types and behavioural prototypes in psychology and psychotherapy.

We study the optimal memorization capacity of modern Hopfield models and Kernelized Hopfield Models (KHMs), a transformer-compatible class of Dense Associative Memories.

However, we now trust that the reviewers are satisfied with the rigour and the correctness of the methodology and the proofs. Therefore, we can drop the proofs of lemmas 4.1 and 4.2 and make the proof of theorem 4.3 more concise so as to have space to expand the introduction to highlight the above points (explained in detail in section 3 below) and add a few words about the replica technique, and include a concluding section.

We summarize our notations in the following table for easy reference.