{"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}
引用次数: 3
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$$ 4-2\sqrt{3} $$ .
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.