{"title":"图的随机子树模型","authors":"Luis Fredes, Jean-Francois Marckert","doi":"10.1214/23-ps22","DOIUrl":null,"url":null,"abstract":"Consider a connected graph G=(E,V) with N=|V| vertices. The main purpose of this paper is to explore the question of uniform sampling of a subtree of G with n nodes, for some n≤N (the spanning tree case correspond to n=N, and is already deeply studied in the literature). We provide new asymptotically exact simulation methods using Markov chains for general connected graphs G, and any n≤N. We highlight the case of the uniform subtree of Z2 with n nodes, containing the origin (0,0) for which Schramm asked several questions. We produce pictures, statistics, and some conjectures. A second aim of the paper is devoted to surveying other models of random subtrees of a graph, among them, DLA models, the first passage percolation, the uniform spanning tree and the minimum spanning tree. We also provide new models, some statistics, and some conjectures.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"Models of random subtrees of a graph\",\"authors\":\"Luis Fredes, Jean-Francois Marckert\",\"doi\":\"10.1214/23-ps22\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Consider a connected graph G=(E,V) with N=|V| vertices. The main purpose of this paper is to explore the question of uniform sampling of a subtree of G with n nodes, for some n≤N (the spanning tree case correspond to n=N, and is already deeply studied in the literature). We provide new asymptotically exact simulation methods using Markov chains for general connected graphs G, and any n≤N. We highlight the case of the uniform subtree of Z2 with n nodes, containing the origin (0,0) for which Schramm asked several questions. We produce pictures, statistics, and some conjectures. A second aim of the paper is devoted to surveying other models of random subtrees of a graph, among them, DLA models, the first passage percolation, the uniform spanning tree and the minimum spanning tree. We also provide new models, some statistics, and some conjectures.\",\"PeriodicalId\":46216,\"journal\":{\"name\":\"Probability Surveys\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.3000,\"publicationDate\":\"2023-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"1\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Probability Surveys\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1214/23-ps22\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Probability Surveys","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1214/23-ps22","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Consider a connected graph G=(E,V) with N=|V| vertices. The main purpose of this paper is to explore the question of uniform sampling of a subtree of G with n nodes, for some n≤N (the spanning tree case correspond to n=N, and is already deeply studied in the literature). We provide new asymptotically exact simulation methods using Markov chains for general connected graphs G, and any n≤N. We highlight the case of the uniform subtree of Z2 with n nodes, containing the origin (0,0) for which Schramm asked several questions. We produce pictures, statistics, and some conjectures. A second aim of the paper is devoted to surveying other models of random subtrees of a graph, among them, DLA models, the first passage percolation, the uniform spanning tree and the minimum spanning tree. We also provide new models, some statistics, and some conjectures.