Some results for range of random walk on graph with spectral dimension two

[1903.07990] Some results for range of random walk on graph with spectral dimension two Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Probability arXiv:1903.07990 (math) [Submitted on 19 Mar 2019 ( v1 ), last revised 1 May 2020 (this version, v2)] Title: Some results for range of random walk on graph with spectral dimension two Authors: Kazuki Okamura View a PDF of the paper titled Some results for range of random walk on graph with spectral dimension two, by Kazuki Okamura View PDF HTML (experimental) Abstract: We consider the range of the simple random walk on graphs with spectral dimension two. We give a form of strong law of large numbers under a certain uniform condition, which is satisfied by not only the square integer lattice but also a class of fractal graphs. Our results imply the strong law of large numbers on the square integer lattice established by Dvoretzky and Erdös (1951). Our proof does not depend on spacial homogeneity of space and gives a new proof of the strong law of large numbers on the lattice. We also show that the behavior of appropriately scaled expectations of the range is stable with respect to every "finite modification" of the two-dimensional integer lattice, and furthermore we construct a recurrent graph such that the uniform condition holds but the scaled expectations fluctuate. As an application, we establish a form of law of the iterated logarithms for lamplighter random walks in the case that the spectral dimension of the underlying graph is two. Comments: 30 pages, minor revision, to appear in Journal of Theoretical Probability Subjects: Probability (math.PR) MSC classes: 60K35 Cite as: arXiv:1903.07990 [math.PR] (or arXiv:1903.07990v2 [math.PR] for this version) https://doi.org/10.48550/arXiv.1903.07990 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Journal of Theoretical Probability 34, No. 3, 1653-1688 (2021) Related DOI : https://doi.org/10.1007/s10959-020-01013-0 Focus to learn more DOI(s) linking to related resources Submission history From: Kazuki Okamura [ view email ] [v1] Tue, 19 Mar 2019 13:26:15 UTC (21 KB) [v2] Fri, 1 May 2020 14:53:07 UTC (22 KB) Full-text links: Access Paper: View a PDF of the paper titled Some results for range of random walk on graph with spectral dimension two, by Kazuki Okamura View PDF HTML (experimental) TeX Source view license Current browse context: math.PR < prev | next > new | recent | 2019-03 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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