Finite size scaling of bitstring probability distributions for Rydberg arrays

We calculate the probabilities $p_{\{n\}}$ of the measured bitstrings $\{n\}$ for the vacuum of Rydberg ladders with $N_q$ atoms. As $N_q$ increases, the $p_{\{n\}}$ decrease but become more dense in the low $p$ region raising the possibility that their smallness could be compensated by their large number. The importance of the low probability states can be estimated from the cumulative probability distribution $\Sigma(p_{\Lambda},N_q)$, which is the probability to observe any state having a probability $p\leq p_{\Lambda}$. For not too large values of $p_{\Lambda}$, it is possible to approximately collapse the $\Sigma(p_{\Lambda},N_q)$ for successive $N_q$ into a function resembling the Fermi function when plotted as a function of $-\ln(p_{\Lambda})$. We show that the number of shots necessary to reduce $\Sigma(p_{\Lambda},N_q)$ to some low enough value grows exponentially with $N_q$. We discuss the implications for calculating observables associated with the vacuum.