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 ℓn and the average number of active users kn may be unbounded. For this channel, we characterize the maximum number of bits that can be transmitted reliably per unit-energy in terms of ℓn and kn. We show that if kn log ℓn is sublinear in n, then each user can achieve the single-user capacity per unit-energy. Conversely, if kn log ℓn 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.