Capacity per Unit-Energy of Gaussian Random Many-Access Channels

We consider 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 be unbounded. For this channel, we characterize the maximum number of bits that can be transmitted reliably per unit-energy in terms of $ell_n$ and $k_n$. We show that if $k_nlog ell_n$ is sublinear in $n$, then each user can achieve the single-user capacity per unit-energy. Conversely, if $k_nlog ell_n$ is superlinear in $n$, then the capacity per unit-energy is zero. We further demonstrate that orthogonal-access schemes, which are optimal when all users are active with probability one, can be strictly suboptimal.