We describe a probabilistic methodology, based on random walk estimates, to obtain exponential upper bounds for the probability of observing unusually small maximal components in two classical (near-)critical random graph models. More specifically, we analyse the near-critical Erd\H{o}s-R\'enyi model $\mathbb{G}(n,p)$ and the random graph $\mathbb{G}(n,d,p)$ obtained by performing near-critical $p$-bond percolation on a simple random $d$-regular graph and show that, for each one of these models, the probability that the size of a largest component is smaller than $n^{2/3}/A$ is at most of order $\exp(-A^{3/2})$. The exponent $3/2$ is known to be optimal for the near-critical $\mathbb{G}(n,p)$ random graph, whereas for the near-critical $\mathbb{G}(n,d,p)$ model the best known upper bound for the above probability was of order $A^{-3/5}$. As a secondary result we show, by means of an optimized version of the martingale method of Nachmias and Peres, that the above probability of observing an unusually small maximal component is at most of order $\exp(-A^{3/5})$ in other two critical models, namely a random intersection graph and the quantum random graph; this stretched-exponential bounds also improve upon the known (polynomial) bounds available for these other two critical models.

翻译：本文提出了一种基于随机游走估计的概率方法，用以获得两个经典（近）临界随机图模型中观测到异常小的最大连通分支概率的指数上界。具体而言，我们分析了近临界Erdős–Rényi模型 $\mathbb{G}(n,p)$ 以及在简单随机 $d$-正则图上进行近临界 $p$-键渗流所获得的随机图 $\mathbb{G}(n,d,p)$，并证明对于这两个模型中的每一个，最大连通分支的尺寸小于 $n^{2/3}/A$ 的概率至多为 $\exp(-A^{3/2})$ 量级。已知指数 $3/2$ 对于近临界 $\mathbb{G}(n,p)$ 随机图是最优的，而对于近临界 $\mathbb{G}(n,d,p)$ 模型，上述概率的已知最佳上界为 $A^{-3/5}$ 量级。作为一项次要结果，我们通过Nachmias和Peres鞅方法的优化版本证明，在其他两个临界模型（即随机交集图和量子随机图）中，观测到异常小的最大连通分支的概率至多为 $\exp(-A^{3/5})$ 量级；该拉伸指数界也改进了这两个其他临界模型的已知（多项式）界。