It is known that many different types of finite random subgraph models undergo quantitatively similar phase transitions around their percolation thresholds, and the proofs of these results rely on isoperimetric properties of the underlying host graph. Recently, the authors showed that such a phase transition occurs in a large class of regular high-dimensional product graphs, generalising a classic result for the hypercube. In this paper we give new isoperimetric inequalities for such regular high-dimensional product graphs, which generalise the well-known isoperimetric inequality of Harper for the hypercube, and are asymptotically sharp for a wide range of set sizes. We then use these isoperimetric properties to investigate the structure of the giant component \(L_1\) in supercritical percolation on these product graphs, that is, when \(p=\frac{1+\epsilon }{d}\), where d is the degree of the product graph and \(\epsilon >0\) is a small enough constant. We show that typically \(L_1\) has edge-expansion \(\Omega \left( \frac{1}{d\ln d}\right) \). Furthermore, we show that \(L_1\) likely contains a linear-sized subgraph with vertex-expansion \(\Omega \left( \frac{1}{d\ln d}\right) \). These results are best possible up to the logarithmic factor in d. Using these likely expansion properties, we determine, up to small polylogarithmic factors in d, the likely diameter of \(L_1\) as well as the typical mixing time of a lazy random walk on \(L_1\). Furthermore, we show the likely existence of a cycle of length \(\Omega \left( \frac{n}{d\ln d}\right) \). These results not only generalise, but also improve substantially upon the known bounds in the case of the hypercube, where in particular the likely diameter and typical mixing time of \(L_1\) were previously only known to be polynomial in d.

众所周知，许多不同类型的有限随机子图模型在其渗流阈值周围经历了定量相似的相变，并且这些结果的证明依赖于基础主图的等周特性。最近，作者表明这种相变发生在一大类常规高维乘积图中，概括了超立方体的经典结果。在本文中，我们为这种规则的高维乘积图给出了新的等周不等式，它推广了众所周知的超立方体哈珀等周不等式，并且对于大范围的集合大小是渐近尖锐的。然后，我们使用这些等周性质来研究这些乘积图上超临界渗流中巨型分量\(L_1\)的结构，即，当\(p=\frac{1+\epsilon }{d}\)时，其中d是乘积图的次数，\(\epsilon >0\)是一个足够小的常数。我们表明，通常\(L_1\)具有边缘扩展\(\Omega \left( \frac{1}{d\ln d}\right) \)。此外，我们表明\(L_1\)可能包含一个具有顶点扩展\(\Omega \left( \frac{1}{d\ln d}\right) \)的线性大小的子图。这些结果最好达到d的对数因子。使用这些可能的扩展属性，我们可以确定d中的小多对数因子、 \(L_1\)的可能直径以及\(L_1\)上惰性随机游走的典型混合时间。此外，我们证明了可能存在长度为\(\Omega \left( \frac{n}{d\ln d}\right) \)的循环。这些结果不仅概括了超立方体的已知界限，而且还大大改进了超立方体的情况，其中特别是\(L_1\)的可能直径和典型混合时间以前只知道是d中的多项式。