  This paper considers a Gaussian multiple-access channel with random user
activity where the total number of users $\ell_n$ and the average number of
active users $k_n$ may grow with the blocklength $n$. For this channel, it
studies the maximum number of bits that can be transmitted reliably per
unit-energy as a function of $\ell_n$ and $k_n$. When all users are active with
probability one, i.e., $\ell_n = k_n$, it is demonstrated that if $k_n$ is of
an order strictly below $n/\log n$, then each user can achieve the single-user
capacity per unit-energy $(\log e)/N_0$ (where $N_0/ 2$ is the noise power) by
using an orthogonal-access scheme. In contrast, if $k_n$ is of an order
strictly above $n/\log n$, then the capacity per unit-energy is zero.
Consequently, there is a sharp transition between orders of growth where
interference-free communication is feasible and orders of growth where reliable
communication at a positive rate per unit-energy is infeasible. It is further
demonstrated that orthogonal-access schemes in combination with orthogonal
codebooks, which achieve the capacity per unit-energy when the number of users
is bounded, can be strictly suboptimal.
  When the user activity is random, i.e., when $\ell_n$ and $k_n$ are
different, it is demonstrated that if $k_n\log \ell_n$ is sublinear in $n$,
then each user can achieve the single-user capacity per unit-energy $(\log
e)/N_0$. Conversely, if $k_n\log \ell_n$ is superlinear in $n$, then the
capacity per unit-energy is zero. Consequently, there is again a sharp
transition between orders of growth where interference-free communication is
feasible and orders of growth where reliable communication at a positive rate
is infeasible that depends on the asymptotic behaviours of both $\ell_n$ and
$k_n$. It is further demonstrated that orthogonal-access schemes, which are
optimal when $\ell_n = k_n$, can be strictly suboptimal.
