Abstract
Position n points uniformly at random in the unit square S, and consider the Voronoi tessellation of S corresponding to the set η of points. Toss a fair coin for each cell in the tessellation to determine whether to colour the cell red or blue. Let denote the event that there exists a red horizontal crossing of S in the resulting colouring. In 1999, Benjamini, Kalai and Schramm conjectured that knowing the tessellation, but not the colouring, asymptotically gives no information as to whether the event will occur or not. More precisely, since occurs with probability , by symmetry, they conjectured that the conditional probabilities converge in probability to 1/2, as . This conjecture was settled in 2016 by Ahlberg, Griffiths, Morris and Tassion. In this paper we derive a stronger bound on the rate at which approaches its mean. As a consequence we strengthen the convergence in probability to almost sure convergence.
Funding Statement
Research in part supported by the Swedish Research Council grant 2016-04442 (DA), FAPERJ bolsa Nota 10 proc. E-26/200.194/2020 and Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – finance code 001 (DdlR), CNPq bolsa de produtividade em pesquisa proc. 307521/2019-2 and FAPERJ Jovem cientista do nosso estado proc. 202.713/2018 (SG).
Citation
Daniel Ahlberg. Daniel de la Riva. Simon Griffiths. "On the rate of convergence in quenched Voronoi percolation." Electron. J. Probab. 26 1 - 26, 2021. https://doi.org/10.1214/21-EJP712
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