Hemispherical Concentration for Semi-Unsourced Random Access in Many-Access Regime

We study a semi-unsourced random access model in which a lightweight coordinator assigns distinct identifiers from a set $\mathcal{D}$ to the active users. Each active user then chooses a message uniformly at random from $\mathcal{M}$, forming an ID-message pair. The users share a spherical codebook whose codewords are drawn independently and uniformly from the hypersphere of radius $\sqrt{nP}$. We analyze the systme in the many-access channel regime, where the number of active users satisfies $K_a(n)=βn+o(n)$, and assume that the total codebook size is $M_n=|\mathcal{D}||\mathcal{M}|=αn+o(n)$. We show that, for $0<β\leq 1$, any $K_a(n)$-subset is almost surely contained in a single hemisphere, and for $1<β<2$, this hemispherical property holds with probability tending to one exponentially. Upon observing the channel output $\mbox{\boldmath $Y$}$, the decoder operates in two steps. In the pre-filtering step, it restricts the sphere to a sequence of spherical caps $\{ \hat{\mathcal{H}}_{n}\}$ converging to the hemisphere with direction $\hat{\mbox{\boldmath $u$}}=\mbox{\boldmath $Y$}/\| \mbox{\boldmath $Y$}\|$ as $n\rightarrow \infty$. In the subsequent maximum likelihood (ML) step, it performs ML estimation over the reduced candidate set. We show that per-user error probability of the pre-filtering step vanishes as $n\rightarrow \infty$, and that the worst-case asymptotic exponential decay rate of the per-user ML error probability over the reduced search space is $P/4$.