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@ARTICLE{Bungert:619275,
      author       = {Bungert, Leon and Calder, Jeff and Roith, Tim},
      title        = {{R}atio convergence rates for {E}uclidean first-passage
                      percolation: {A}pplications to the graph infinity
                      {L}aplacian},
      journal      = {The annals of applied probability},
      volume       = {34},
      number       = {4},
      issn         = {1050-5164},
      address      = {Hayward, Calif.},
      publisher    = {Project Euclid},
      reportid     = {PUBDB-2024-07523},
      pages        = {3870 - 3910},
      year         = {2024},
      abstract     = {n this paper we prove the first quantitative convergence
                      rates for the graph infinity Laplace equation for length
                      scales at the connectivity threshold. In the graph-based
                      semisupervised learning community this equation is also
                      known as Lipschitz learning. The graph infinity Laplace
                      equation is characterized by the metric on the underlying
                      space, and convergence rates follow from convergence rates
                      for graph distances. At the connectivity threshold, this
                      problem is related to Euclidean first passage percolation,
                      which is concerned with the Euclidean distance function
                      d$_h$(x,y) on a homogeneous Poisson point process on
                      $\mathbb{R}_d$, where admissible paths have step size at
                      most h>0. Using a suitable regularization of the distance
                      function and subadditivity we prove that
                      $d_{h_s}$(0,se$_1$)/s→σ as s→∞ almost surely where
                      σ≥1 is a dimensional constant and h$_s$≳log(s)$^{1/d}$.
                      A convergence rate is not available due to a lack of
                      approximate superadditivity when h$_s$→∞. Instead, we
                      prove convergence rates for the ratio
                      $\frac{dh(0,se1)}{dh(0,2se1)}$→$\frac{1}{2}$ when h is
                      frozen and does not depend on s. Combining this with the
                      techniques that we developed in (IMA J. Numer. Anal. 43
                      (2023) 2445–2495), we show that this notion of ratio
                      convergence is sufficient to establish uniform convergence
                      rates for solutions of the graph infinity Laplace equation
                      at percolation length scales.},
      cin          = {FS-CI},
      ddc          = {510},
      cid          = {I:(DE-H253)FS-CI-20230420},
      pnm          = {623 - Data Management and Analysis (POF4-623) / DFG project
                      G:(GEPRIS)390685813 - EXC 2047: Hausdorff Center for
                      Mathematics (HCM) (390685813) / NoMADS - Nonlocal Methods
                      for Arbitrary Data Sources (777826)},
      pid          = {G:(DE-HGF)POF4-623 / G:(GEPRIS)390685813 /
                      G:(EU-Grant)777826},
      experiment   = {EXP:(DE-MLZ)NOSPEC-20140101},
      typ          = {PUB:(DE-HGF)16},
      UT           = {WOS:001288337500011},
      doi          = {10.1214/24-AAP2052},
      url          = {https://bib-pubdb1.desy.de/record/619275},
}