{"title":"1‐独立的渗流作用在0 2×Kn上","authors":"Victor Falgas‐Ravry, Vincent Pfenninger","doi":"10.1002/rsa.21129","DOIUrl":null,"url":null,"abstract":"A random graph model on a host graph H$$ H $$ is said to be 1‐independent if for every pair of vertex‐disjoint subsets A,B$$ A,B $$ of E(H)$$ E(H) $$ , the state of edges (absent or present) in A$$ A $$ is independent of the state of edges in B$$ B $$ . For an infinite connected graph H$$ H $$ , the 1‐independent critical percolation probability p1,c(H)$$ {p}_{1,c}(H) $$ is the infimum of the p∈[0,1]$$ p\\in \\left[0,1\\right] $$ such that every 1‐independent random graph model on H$$ H $$ in which each edge is present with probability at least p$$ p $$ almost surely contains an infinite connected component. Balister and Bollobás observed in 2012 that p1,c(ℤd)$$ {p}_{1,c}\\left({\\mathbb{Z}}^d\\right) $$ tends to a limit in [12,1]$$ \\left[\\frac{1}{2},1\\right] $$ as d→∞$$ d\\to \\infty $$ , and they asked for the value of this limit. We make progress on a related problem by showing that limn→∞p1,c(ℤ2×Kn)=4−23=0.5358….$$ \\underset{n\\to \\infty }{\\lim }{p}_{1,c}\\left({\\mathbb{Z}}^2\\times {K}_n\\right)=4-2\\sqrt{3}=0.5358\\dots . $$In fact, we show that the equality above remains true if the sequence of complete graphs Kn$$ {K}_n $$ is replaced by a sequence of weakly pseudorandom graphs on n$$ n $$ vertices with average degree ω(logn)$$ \\omega \\left(\\log n\\right) $$ . We conjecture the answer to Balister and Bollobás's question is also 4−23$$ 4-2\\sqrt{3} $$ .","PeriodicalId":54523,"journal":{"name":"Random Structures & Algorithms","volume":"104 1","pages":"887 - 910"},"PeriodicalIF":0.9000,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"3","resultStr":"{\"title\":\"1‐independent percolation on ℤ2×Kn\",\"authors\":\"Victor Falgas‐Ravry, Vincent Pfenninger\",\"doi\":\"10.1002/rsa.21129\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"A random graph model on a host graph H$$ H $$ is said to be 1‐independent if for every pair of vertex‐disjoint subsets A,B$$ A,B $$ of E(H)$$ E(H) $$ , the state of edges (absent or present) in A$$ A $$ is independent of the state of edges in B$$ B $$ . For an infinite connected graph H$$ H $$ , the 1‐independent critical percolation probability p1,c(H)$$ {p}_{1,c}(H) $$ is the infimum of the p∈[0,1]$$ p\\\\in \\\\left[0,1\\\\right] $$ such that every 1‐independent random graph model on H$$ H $$ in which each edge is present with probability at least p$$ p $$ almost surely contains an infinite connected component. Balister and Bollobás observed in 2012 that p1,c(ℤd)$$ {p}_{1,c}\\\\left({\\\\mathbb{Z}}^d\\\\right) $$ tends to a limit in [12,1]$$ \\\\left[\\\\frac{1}{2},1\\\\right] $$ as d→∞$$ d\\\\to \\\\infty $$ , and they asked for the value of this limit. We make progress on a related problem by showing that limn→∞p1,c(ℤ2×Kn)=4−23=0.5358….$$ \\\\underset{n\\\\to \\\\infty }{\\\\lim }{p}_{1,c}\\\\left({\\\\mathbb{Z}}^2\\\\times {K}_n\\\\right)=4-2\\\\sqrt{3}=0.5358\\\\dots . $$In fact, we show that the equality above remains true if the sequence of complete graphs Kn$$ {K}_n $$ is replaced by a sequence of weakly pseudorandom graphs on n$$ n $$ vertices with average degree ω(logn)$$ \\\\omega \\\\left(\\\\log n\\\\right) $$ . We conjecture the answer to Balister and Bollobás's question is also 4−23$$ 4-2\\\\sqrt{3} $$ .\",\"PeriodicalId\":54523,\"journal\":{\"name\":\"Random Structures & Algorithms\",\"volume\":\"104 1\",\"pages\":\"887 - 910\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2022-12-21\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"3\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Random Structures & Algorithms\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1002/rsa.21129\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, SOFTWARE ENGINEERING\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Random Structures & Algorithms","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1002/rsa.21129","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 3
摘要
如果对于E(H) $$ E(H) $$的每一对顶点不相交的子集A,B $$ A,B $$, A $$ A $$中的边的状态(不存在或存在)与B $$ B $$中的边的状态无关,则表示主机图H $$ H $$上的随机图模型是1独立的。对于无限连通图H $$ H $$,1独立的临界渗透概率p1,c(H) $$ {p}_{1,c}(H) $$是p∈[0,1]$$ p\in \left[0,1\right] $$的下限值,使得H $$ H $$上每条边至少以p $$ p $$的概率出现的每一个1独立随机图模型几乎肯定包含一个无限连通分量。Balister和Bollobás在2012年观察到,当d→∞$$ d\to \infty $$时,p1,c(lgd) $$ {p}_{1,c}\left({\mathbb{Z}}^d\right) $$趋向于[12,1]$$ \left[\frac{1}{2},1\right] $$中的一个极限,他们要求这个极限的值。通过证明limn→∞p1,c(0 2×Kn)=4−23=0.5358....,我们在一个相关问题上取得了进展$$ \underset{n\to \infty }{\lim }{p}_{1,c}\left({\mathbb{Z}}^2\times {K}_n\right)=4-2\sqrt{3}=0.5358\dots . $$事实上,我们证明了如果完全图序列Kn $$ {K}_n $$被n个$$ n $$顶点上的平均度为ω(logn) $$ \omega \left(\log n\right) $$的弱伪随机图序列所取代,上面的等式仍然成立。我们推测Balister和Bollobás的问题的答案也是4−23 $$ 4-2\sqrt{3} $$。
A random graph model on a host graph H$$ H $$ is said to be 1‐independent if for every pair of vertex‐disjoint subsets A,B$$ A,B $$ of E(H)$$ E(H) $$ , the state of edges (absent or present) in A$$ A $$ is independent of the state of edges in B$$ B $$ . For an infinite connected graph H$$ H $$ , the 1‐independent critical percolation probability p1,c(H)$$ {p}_{1,c}(H) $$ is the infimum of the p∈[0,1]$$ p\in \left[0,1\right] $$ such that every 1‐independent random graph model on H$$ H $$ in which each edge is present with probability at least p$$ p $$ almost surely contains an infinite connected component. Balister and Bollobás observed in 2012 that p1,c(ℤd)$$ {p}_{1,c}\left({\mathbb{Z}}^d\right) $$ tends to a limit in [12,1]$$ \left[\frac{1}{2},1\right] $$ as d→∞$$ d\to \infty $$ , and they asked for the value of this limit. We make progress on a related problem by showing that limn→∞p1,c(ℤ2×Kn)=4−23=0.5358….$$ \underset{n\to \infty }{\lim }{p}_{1,c}\left({\mathbb{Z}}^2\times {K}_n\right)=4-2\sqrt{3}=0.5358\dots . $$In fact, we show that the equality above remains true if the sequence of complete graphs Kn$$ {K}_n $$ is replaced by a sequence of weakly pseudorandom graphs on n$$ n $$ vertices with average degree ω(logn)$$ \omega \left(\log n\right) $$ . We conjecture the answer to Balister and Bollobás's question is also 4−23$$ 4-2\sqrt{3} $$ .
期刊介绍:
It is the aim of this journal to meet two main objectives: to cover the latest research on discrete random structures, and to present applications of such research to problems in combinatorics and computer science. The goal is to provide a natural home for a significant body of current research, and a useful forum for ideas on future studies in randomness.
Results concerning random graphs, hypergraphs, matroids, trees, mappings, permutations, matrices, sets and orders, as well as stochastic graph processes and networks are presented with particular emphasis on the use of probabilistic methods in combinatorics as developed by Paul Erdõs. The journal focuses on probabilistic algorithms, average case analysis of deterministic algorithms, and applications of probabilistic methods to cryptography, data structures, searching and sorting. The journal also devotes space to such areas of probability theory as percolation, random walks and combinatorial aspects of probability.