{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/phase-transition-for-the-late-points-of","title":"Phase transition for the late points of random walk","arxiv_id":"2309.03192","date":"2023-09-06","proceeding":null,"authors":["Alexis Prévost","Pierre-François Rodriguez","Perla Sousi"],"abstract":"Let $X$ be a random walk on the torus of side length $N$ in dimension $d\\geq 3$ with uniform starting point, and $t_{\\text{cov}}$ be the expected value of its cover time, which is the first time that $X$ has visited every vertex of the torus at least once. For $\\alpha > 0$, the set $\\mathcal{L}^{\\alpha}$ of $\\alpha$-late points consists of those points not visited by $X$ at time $\\alpha t_{\\text{cov}}$. We prove the existence of a value $\\alpha_* \\in (\\frac12,1)$ across which $\\mathcal{L}^{\\alpha}$ trivialises as follows: for all $\\alpha > \\alpha_*$ and $\\epsilon\\geq N^{-c}$ there exists a coupling of $\\mathcal{L}^\\alpha$ and two occupation sets $\\mathcal{B}^{\\alpha_\\pm}$ of i.i.d. Bernoulli fields having the same density as $\\mathcal{L}^{\\alpha\\pm \\epsilon}$, which is asymptotic to $N^{-(\\alpha\\pm\\epsilon)d}$, with the property that the inclusion $ \\mathcal{B}^{\\alpha_+} \\subseteq \\mathcal{L}^{\\alpha} \\subseteq \\mathcal{B}^{\\alpha_-}$ holds with high probability as $N \\to \\infty$. On the contrary, when $\\alpha \\leq \\alpha_*$ there is no such coupling. Corresponding results also hold for the vacant set of random interlacements at high intensities. The transition at $\\alpha_*$ corresponds to the (dis-)appearance of `double-points' (i.e. neighboring pairs of points) in $\\mathcal{L}^\\alpha$. We further describe the law of $\\mathcal{L}^{\\alpha}$ for $\\alpha>\\frac12$ by adding independent patterns to $\\mathcal{B}^{\\alpha_{\\pm}}$. In dimensions $d \\geq 4$ these are exactly all two-point sets. When $d=3$ one must also include all connected three-point sets, but no other.","url_abs":"https://arxiv.org/abs/2309.03192v1","url_pdf":"https://arxiv.org/pdf/2309.03192v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"phase-transition-for-the-late-points-of","repo_url":"https://github.com/a-prevost/capacity","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}