{"title":"临界高尔顿-沃森树上线性边增强随机漫步的缩放极限","authors":"G. Andriopoulos, Eleanor Archer","doi":"10.1214/23-ejp901","DOIUrl":null,"url":null,"abstract":"We prove an invariance principle for linearly edge reinforced random walks on $\\gamma$-stable critical Galton-Watson trees, where $\\gamma \\in (1,2]$ and where the edge joining $x$ to its parent has rescaled initial weight $d(\\rho, x)^{\\alpha}$ for some $\\alpha \\leq 1$. This corresponds to the recurrent regime of initial weights. We then establish fine asymptotics for the limit process. In the transient regime, we also give an upper bound on the random walk displacement in the discrete setting, showing that the edge reinforced random walk never has positive speed, even when the initial edge weights are strongly biased away from the root.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":" ","pages":""},"PeriodicalIF":1.1000,"publicationDate":"2021-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees\",\"authors\":\"G. Andriopoulos, Eleanor Archer\",\"doi\":\"10.1214/23-ejp901\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove an invariance principle for linearly edge reinforced random walks on $\\\\gamma$-stable critical Galton-Watson trees, where $\\\\gamma \\\\in (1,2]$ and where the edge joining $x$ to its parent has rescaled initial weight $d(\\\\rho, x)^{\\\\alpha}$ for some $\\\\alpha \\\\leq 1$. This corresponds to the recurrent regime of initial weights. We then establish fine asymptotics for the limit process. In the transient regime, we also give an upper bound on the random walk displacement in the discrete setting, showing that the edge reinforced random walk never has positive speed, even when the initial edge weights are strongly biased away from the root.\",\"PeriodicalId\":50538,\"journal\":{\"name\":\"Electronic Journal of Probability\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":1.1000,\"publicationDate\":\"2021-12-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Electronic Journal of Probability\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1214/23-ejp901\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Journal of Probability","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1214/23-ejp901","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
Scaling limit of linearly edge-reinforced random walks on critical Galton-Watson trees
We prove an invariance principle for linearly edge reinforced random walks on $\gamma$-stable critical Galton-Watson trees, where $\gamma \in (1,2]$ and where the edge joining $x$ to its parent has rescaled initial weight $d(\rho, x)^{\alpha}$ for some $\alpha \leq 1$. This corresponds to the recurrent regime of initial weights. We then establish fine asymptotics for the limit process. In the transient regime, we also give an upper bound on the random walk displacement in the discrete setting, showing that the edge reinforced random walk never has positive speed, even when the initial edge weights are strongly biased away from the root.
期刊介绍:
The Electronic Journal of Probability publishes full-size research articles in probability theory. The Electronic Communications in Probability (ECP), a sister journal of EJP, publishes short notes and research announcements in probability theory.
Both ECP and EJP are official journals of the Institute of Mathematical Statistics
and the Bernoulli society.