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@ARTICLE{Grassberger:840250,
      author       = {Grassberger, Peter},
      title        = {{H}ow fast does a random walk cover a torus?},
      journal      = {Physical review / E},
      volume       = {96},
      number       = {1},
      issn         = {2470-0045},
      address      = {Woodbury, NY},
      publisher    = {Inst.},
      reportid     = {FZJ-2017-07803},
      pages        = {012115},
      year         = {2017},
      abstract     = {We present high statistics simulation data for the average
                      time ⟨Tcover(L)⟩ that a random walk needs to cover
                      completely a two-dimensional torus of size L×L. They
                      confirm the mathematical prediction that
                      ⟨Tcover(L)⟩∼(LlnL)2 for large L, but the prefactor
                      seems to deviate significantly from the supposedly exact
                      result 4/π derived by Dembo et al. [Ann. Math. 160, 433
                      (2004)], if the most straightforward extrapolation is used.
                      On the other hand, we find that this scaling does hold for
                      the time TN(t)=1(L) at which the average number of yet
                      unvisited sites is 1, as also predicted previously. This
                      might suggest (wrongly) that ⟨Tcover(L)⟩ and TN(t)=1(L)
                      scale differently, although the distribution of rescaled
                      cover times becomes sharp in the limit L→∞. But our
                      results can be reconciled with those of Dembo et al. by a
                      very slow and nonmonotonic convergence of
                      ⟨Tcover(L)⟩/(LlnL)2, as had been indeed proven by Belius
                      et al. [Probab. Theory Relat. Fields 167, 461 (2017)] for
                      Brownian walks, and was conjectured by them to hold also for
                      lattice walks.},
      cin          = {JSC},
      ddc          = {530},
      cid          = {I:(DE-Juel1)JSC-20090406},
      pnm          = {511 - Computational Science and Mathematical Methods
                      (POF3-511)},
      pid          = {G:(DE-HGF)POF3-511},
      typ          = {PUB:(DE-HGF)16},
      pubmed       = {pmid:29347167},
      UT           = {WOS:000405194200005},
      doi          = {10.1103/PhysRevE.96.012115},
      url          = {https://juser.fz-juelich.de/record/840250},
}