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Sharpness of the Phase Transition for the Orthant Model

Abstract

The orthant model is a directed percolation model on \(\mathbb {Z}^{d}\), in which all clusters are infinite. We prove a sharp threshold result for this model: if p is larger than the critical value above which the cluster of 0 is contained in a cone, then the shift from 0 that is required to contain the cluster of 0 in that cone is exponentially small. As a consequence, above this critical threshold, a shape theorem holds for the cluster of 0, as well as ballisticity of the random walk on this cluster.

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Acknowledgments

I thank Matija Pasch for insightful discussions on the topic, as well as for useful comments on the manuscript.

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Open Access funding enabled and organized by Projekt DEAL.

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Correspondence to Thomas Beekenkamp.

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Beekenkamp, T. Sharpness of the Phase Transition for the Orthant Model. Math Phys Anal Geom 24, 36 (2021). https://doi.org/10.1007/s11040-021-09408-z

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Keywords

  • Oriented percolation
  • Phase transitions
  • Degenerate random environments
  • Random graphs

Mathematics Subject Classification (2010)

  • 60K35