Asymptotic degeneracies of M2-brane SCFTs

We study the asymptotic growth of the degeneracy of the BPS local operators with scaling dimension $n\mathrm{/}2$ in the three-dimensional superconformal field theories describing $N$ M2-branes. From the large $N$ supersymmetric indices we obtain the asymptotic formulas for degeneracies of the M2-brane SCFTs according to the Meinardus theorem. We observe an intriguing universal $n^{2\mathrm{/}3}$ growth of the degeneracies in various theories of M2-brane SCFTs. We also determine the coefficients of $n^{2\mathrm{/}3}$ growth as well as further corrections in these theories explicitly.