First order distinguishability of sparse random graphs
<|reference_start|>First order distinguishability of sparse random graphs: We study the problem of distinguishing between two independent samples $\mathbf{G}_n^1,\mathbf{G}_n^2$ of a binomial random graph $G(n,p)$ by first order (FO) sentences. Shelah and Spencer proved that, for a constant $\alpha\in(0,1)$, $G(n,n^{-\alpha})$ obeys FO zero-one law if and only if $\alpha$ is irrational. Therefore, for irrational $\alpha\in(0,1)$, any fixed FO sentence does not distinguish between $\mathbf{G}_n^1,\mathbf{G}_n^2$ with asymptotical probability 1 (w.h.p.) as $n\to\infty$. We show that the minimum quantifier depth $\mathbf{k}_{\alpha}$ of a FO sentence $\varphi=\varphi(\mathbf{G}_n^1,\mathbf{G}_n^2)$ distinguishing between $\mathbf{G}_n^1,\mathbf{G}_n^2$ depends on how closely $\alpha$ can be approximated by rationals: (1) for all non-Liouville $\alpha\in(0,1)$, $\mathbf{k}_{\alpha}=\Omega(\ln\ln\ln n)$ w.h.p.; (2) there are irrational $\alpha\in(0,1)$ with $\mathbf{k}_{\alpha}$ that grow arbitrarily slowly w.h.p.; (3) $\mathbf{k}_{\alpha}=O_p(\frac{\ln n}{\ln\ln n})$ for all $\alpha\in(0,1)$. The main ingredients in our proofs are a novel randomized algorithm that generates asymmetric strictly balanced graphs as well as a new method to study symmetry groups of randomly perturbed graphs.<|reference_end|>
@article{hershko2024first,
    title={First order distinguishability of sparse random graphs},
    author={Tal Hershko, Maksim Zhukovskii},
    journal={arXiv preprint arXiv:2405.09146},
    year={2024},
    archivePrefix={arXiv},
    eprint={2405.09146},
    primaryClass={math.CO cs.DM cs.LO}
}