{"title":"临界渗流勘探过程在k-OUT图上的收敛性","authors":"Y. Ota","doi":"10.18910/57663","DOIUrl":null,"url":null,"abstract":"We consider the percolation on the k-out graph Gout(n, k). The critical probability of it is 1 k+ √ k2−k . Similarly to the random graph G(n, p), in a scaling window 1 k+ √ k2−k ( 1 + O(n−1/3) ) , the sequence of sizes of large components rescaled by n−2/3 converges to the excursion lengths of a Brownian motion with some drift. Also, the size of the largest component is O(log n) in the subcritical phase, and O(n) in the supercritical phase. The proof is based on the analysis of the exploration process.","PeriodicalId":54660,"journal":{"name":"Osaka Journal of Mathematics","volume":"52 1","pages":"677-719"},"PeriodicalIF":0.5000,"publicationDate":"2015-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"THE CONVERGENCE OF THE EXPLORATION PROCESS FOR CRITICAL PERCOLATION ON THE k-OUT GRAPH\",\"authors\":\"Y. Ota\",\"doi\":\"10.18910/57663\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We consider the percolation on the k-out graph Gout(n, k). The critical probability of it is 1 k+ √ k2−k . Similarly to the random graph G(n, p), in a scaling window 1 k+ √ k2−k ( 1 + O(n−1/3) ) , the sequence of sizes of large components rescaled by n−2/3 converges to the excursion lengths of a Brownian motion with some drift. Also, the size of the largest component is O(log n) in the subcritical phase, and O(n) in the supercritical phase. The proof is based on the analysis of the exploration process.\",\"PeriodicalId\":54660,\"journal\":{\"name\":\"Osaka Journal of Mathematics\",\"volume\":\"52 1\",\"pages\":\"677-719\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2015-07-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Osaka Journal of Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.18910/57663\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Osaka Journal of Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.18910/57663","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
THE CONVERGENCE OF THE EXPLORATION PROCESS FOR CRITICAL PERCOLATION ON THE k-OUT GRAPH
We consider the percolation on the k-out graph Gout(n, k). The critical probability of it is 1 k+ √ k2−k . Similarly to the random graph G(n, p), in a scaling window 1 k+ √ k2−k ( 1 + O(n−1/3) ) , the sequence of sizes of large components rescaled by n−2/3 converges to the excursion lengths of a Brownian motion with some drift. Also, the size of the largest component is O(log n) in the subcritical phase, and O(n) in the supercritical phase. The proof is based on the analysis of the exploration process.
期刊介绍：
Osaka Journal of Mathematics is published quarterly by the joint editorship of the Department of Mathematics, Graduate School of Science, Osaka University, and the Department of Mathematics, Faculty of Science, Osaka City University and the Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University with the cooperation of the Department of Mathematical Sciences, Faculty of Engineering Science, Osaka University. The Journal is devoted entirely to the publication of original works in pure and applied mathematics.
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