Isoperimetric Inequalities and Supercritical Percolation on High-Dimensional Graphs

IF 1 2区数学 Q1 MATHEMATICS

Combinatorica Pub Date : 2024-04-04 DOI:10.1007/s00493-024-00089-0

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引用次数: 0

Abstract

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.

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高维图上的等周不等式和超临界渗流

摘要众所周知，许多不同类型的有限随机子图模型在其渗流阈值附近会发生数量上相似的相变，而这些结果的证明依赖于底层主图的等周特性。最近，作者证明了这种相变发生在一大类规则的高维积图中，推广了超立方体的经典结果。在本文中，我们为这类正则高维积图给出了新的等周不等式，这些等周不等式概括了著名的 Harper 关于超立方体的等周不等式，并且在很宽的集合大小范围内都是渐近尖锐的。然后，我们利用这些等周性质来研究这些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 是积图的度，而 \(\epsilon>0\)是一个足够小的常数。我们证明了典型的 \(L_1\) 有边扩展 \(\Omega \left( \frac{1}{d\ln d}\right) \) 。此外，我们证明了 \(L_1\) 很可能包含一个线性大小的子图，其顶点展开为 \(\Omega \left(\frac{1}{d\ln d}\right) \)。利用这些可能的扩展特性，我们确定了（直到 d 的小对数因子）\(L_1\) 的可能直径以及懒惰随机行走在 \(L_1\) 上的典型混合时间。此外，我们还证明了一个长度为 \(ω \left( \frac{n}{d\ln d}\right) \)的循环的可能存在性。这些结果不仅概括了超立方体的情况，而且大大改进了超立方体的已知边界，特别是在\(L_1\)的可能直径和典型混合时间方面，以前只知道是 d 的多项式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Combinatorica 数学-数学

CiteScore

1.90

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： COMBINATORICA publishes research papers in English in a variety of areas of combinatorics and the theory of computing, with particular emphasis on general techniques and unifying principles. Typical but not exclusive topics covered by COMBINATORICA are - Combinatorial structures (graphs, hypergraphs, matroids, designs, permutation groups). - Combinatorial optimization. - Combinatorial aspects of geometry and number theory. - Algorithms in combinatorics and related fields. - Computational complexity theory. - Randomization and explicit construction in combinatorics and algorithms.