  Recent generalizations of the Hopfield model of associative memories are able
to store a number $P$ of random patterns that grows exponentially with the
number $N$ of neurons, $P=\exp(\alpha N)$. Besides the huge storage capacity,
another interesting feature of these networks is their connection to the
attention mechanism which is part of the Transformer architectures widely
applied in deep learning. In this work, we study a generic family of pattern
ensembles using a statistical mechanics analysis which gives exact asymptotic
thresholds for the retrieval of a typical pattern, $\alpha_1$, and lower bounds
for the maximum of the load $\alpha$ for which all patterns can be retrieved,
$\alpha_c$, as well as sizes of attraction basins. We discuss in detail the
cases of Gaussian and spherical patterns, and show that they display rich and
qualitatively different phase diagrams.
