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.

翻译：我们研究通过一阶（FO）语句区分二项随机图$G(n,p)$的两个独立样本$\mathbf{G}_n^1,\mathbf{G}_n^2$的问题。Shelah与Spencer证明，对于常数$\alpha\in(0,1)$，图$G(n,n^{-\alpha})$服从FO零一律当且仅当$\alpha$为无理数。因此，对于无理数$\alpha\in(0,1)$，任意固定FO语句均不能以渐近概率1（w.h.p.）在$n\to\infty$时区分$\mathbf{G}_n^1,\mathbf{G}_n^2$。本文表明，区分$\mathbf{G}_n^1,\mathbf{G}_n^2$的FO语句$\varphi=\varphi(\mathbf{G}_n^1,\mathbf{G}_n^2)$的最小量词深度$\mathbf{k}_{\alpha}$取决于$\alpha$可被有理数逼近的精度：(1) 对于所有非Liouville数$\alpha\in(0,1)$，$\mathbf{k}_{\alpha}=\Omega(\ln\ln\ln n)$ w.h.p.；(2) 存在无理数$\alpha\in(0,1)$使得$\mathbf{k}_{\alpha}$以任意慢速度增长 w.h.p.；(3) 对所有$\alpha\in(0,1)$，$\mathbf{k}_{\alpha}=O_p(\frac{\ln n}{\ln\ln n})$。我们证明的主要工具包括一种生成非对称严格平衡图的新型随机化算法，以及研究随机扰动图对称群的新方法。