# Scaling Laws for Gaussian Random Many-Access Channels

This paper considers a Gaussian multiple-access channel with random user activity where the total number of users ℓ_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 ℓ_n and k_n. When all users are active with probability one, i.e., ℓ_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 users cannot achieve any positive rate per unit-energy. 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 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 ℓ_n and k_n are different, it is demonstrated that if k_nlogℓ_n is sublinear in n, then each user can achieve the single-user capacity per unit-energy (log e)/N_0. Conversely, if k_nlogℓ_n is superlinear in n, then the users cannot achieve any positive rate per unit-energy. 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 ℓ_n and k_n. It is further demonstrated that orthogonal-access schemes, which are optimal when ℓ_n = k_n, can be strictly suboptimal in general.

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