{"title":"Critical window for the vacant set left by random walk on the configuration model","authors":"J. Černý, T. Hayder","doi":"10.30757/alea.v19-10","DOIUrl":null,"url":null,"abstract":"We study the simple random walk on the configuration model with given degree sequence (d1 , . . . , d n n) and investigate the connected components of its vacant set at level u > 0. We show that the size of the maximal connected component exhibits a phase transition at level u∗ which can be related with the critical parameter of random interlacements on a certain Galton-Watson tree. We further show that there is a critical window of size n−1/3 around u∗ in which the largest connected components of the vacant set have a metric space scaling limit resembling the one of the critical Erdős-Rényi random graph.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":"66 1","pages":""},"PeriodicalIF":0.6000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Alea-Latin American Journal of Probability and Mathematical Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.30757/alea.v19-10","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 1
Abstract
We study the simple random walk on the configuration model with given degree sequence (d1 , . . . , d n n) and investigate the connected components of its vacant set at level u > 0. We show that the size of the maximal connected component exhibits a phase transition at level u∗ which can be related with the critical parameter of random interlacements on a certain Galton-Watson tree. We further show that there is a critical window of size n−1/3 around u∗ in which the largest connected components of the vacant set have a metric space scaling limit resembling the one of the critical Erdős-Rényi random graph.
期刊介绍:
ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted.
ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper.
ALEA is affiliated with the Institute of Mathematical Statistics.