VSE knjižnice (vzajemna bibliografsko-kataložna baza podatkov COBIB.SI)
  • Slow convergence in bootstrap percolation
    Gravner, Janko ; Holroyd, Alexander E.
    In the bootstrap percolation model, sites in an ▫$L \times L$▫ square are initially infected independently with probability ▫$p$▫. At subsequent steps, a healthy site becomes infected if it has at ... least two infected neighbors. As ▫$(L,p) \to (\infty,0$▫, the probability that the entire square is eventually infected is known to undergo a phase transition in the parameter ▫$p\log L$▫, occurring asymptotically at ▫$\lambda = \pi^2/18$▫ [Probab. Theory Related Fields 125 (2003) 195-224]. We prove that the discrepancy between the critical parameter and its limit ▫$\lambda$▫ is at least ▫$\Omega((\log L)^{-1/2})$▫. In contrast, the critical window has width only ▫$\Theta((\log L)^{-1})$▫. For the so-called modified model, we prove rigorous explicit bounds which imply, for example, that the relative discrepancy is at least ▫$1\%$▫ even when ▫$L=10^{3000}$▫. Our results shed some light on the observed differences between simulations and rigorous asymptotics.
    Vir: Annals of applied probability. - ISSN 1050-5164 (Vol. 18, no. 3, 2008, str. 909-928)
    Vrsta gradiva - članek, sestavni del
    Leto - 2008
    Jezik - angleški
    COBISS.SI-ID - 14773849

    Povezava(-e):

    http://dx.doi.org/10.1214/07-AAP473

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