{"title":"Isoperimetric Inequalities and Supercritical Percolation on High-Dimensional Graphs","authors":"","doi":"10.1007/s00493-024-00089-0","DOIUrl":null,"url":null,"abstract":"<h3>Abstract</h3> <p>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 <span> <span>\\(L_1\\)</span> </span> in supercritical percolation on these product graphs, that is, when <span> <span>\\(p=\\frac{1+\\epsilon }{d}\\)</span> </span>, where <em>d</em> is the degree of the product graph and <span> <span>\\(\\epsilon >0\\)</span> </span> is a small enough constant. We show that typically <span> <span>\\(L_1\\)</span> </span> has edge-expansion <span> <span>\\(\\Omega \\left( \\frac{1}{d\\ln d}\\right) \\)</span> </span>. Furthermore, we show that <span> <span>\\(L_1\\)</span> </span> likely contains a linear-sized subgraph with vertex-expansion <span> <span>\\(\\Omega \\left( \\frac{1}{d\\ln d}\\right) \\)</span> </span>. These results are best possible up to the logarithmic factor in <em>d</em>. Using these likely expansion properties, we determine, up to small polylogarithmic factors in <em>d</em>, the likely diameter of <span> <span>\\(L_1\\)</span> </span> as well as the typical mixing time of a lazy random walk on <span> <span>\\(L_1\\)</span> </span>. Furthermore, we show the likely existence of a cycle of length <span> <span>\\(\\Omega \\left( \\frac{n}{d\\ln d}\\right) \\)</span> </span>. 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 <span> <span>\\(L_1\\)</span> </span> were previously only known to be polynomial in <em>d</em>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"54 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Combinatorica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00493-024-00089-0","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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.
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.
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