We consider supercritical bond percolation on a family of high‐girth ‐regular expanders. The previous study of Alon, Benjamini and Stacey established that its critical probability for the appearance of a linear‐sized (“giant”) component is . Our main result recovers the sharp asymptotics of the size and degree distribution of the vertices in the giant and its 2‐core at any . It was further shown in the previous study that the second largest component, at any , has size at most for some . We show that, unlike the situation in the classical Erdős‐Rényi random graph, the second largest component in bond percolation on a regular expander, even with an arbitrarily large girth, can have size for arbitrarily close to 1. Moreover, as a by‐product of that construction, we answer negatively a question of Benjamini on the relation between the diameter of a component in percolation on expanders and the existence of a giant component. Finally, we establish other typical features of the giant component, for example, the existence of a linear path.

中文翻译：

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高周长扩张器的渐近性

我们考虑在高周长常规膨胀剂系列上的超临界键渗滤。对Alon，Benjamini和Stacey的先前研究确定，其出现线性大小（“巨型”）分量的临界概率为。我们的主要结果恢复了巨人及其任何2核中顶点的大小和程度分布的清晰渐近性。在先前的研究中进一步表明，第二最大的组件在某些情况下最多具有大小。我们证明了，不像传统的鄂尔多斯-莱利随机图的情况下，债券渗流定期扩展的第二大组成部分，即使任意大的周长，可以有大小为任意接近于1。此外，作为该构造的副产品，我们否定了Benjamini问题，即膨胀机上渗滤中组分的直径与巨型组分的存在之间的关系。最后，我们建立了巨型组件的其他典型特征，例如，线性路径的存在。