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The box-crossing property for critical two-dimensional oriented percolation

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Abstract

We consider critical oriented Bernoulli percolation on the square lattice \(\mathbb {Z}^2\). We prove a Russo–Seymour–Welsh type result which allows us to derive several new results concerning the critical behavior:

  • We establish that the probability that the origin is connected to distance n decays polynomially fast in n.

  • We prove that the critical cluster of 0 conditioned to survive to distance n has a typical width \(w_n\) satisfying \(\varepsilon n^{2/5}\le w_n\le n^{1-\varepsilon }\) for some \(\varepsilon >0\).

The sub-linear polynomial fluctuations contrast with the supercritical regime where \(w_n\) is known to behave linearly in n. It is also different from the critical picture obtained for non-oriented Bernoulli percolation, in which the scaling limit is non-degenerate in both directions. All our results extend to the graphical representation of the one-dimensional contact process.

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Notes

  1. An event E is increasing if it is stable with respect to opening edges.

  2. Formally, this can be seen as the largest left–right crossing of \(B_r\) for the natural lexicographical order on path induced by the lexicographical order on vertices and the order that the edge going left from a vertex is smaller than the edge going right.

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Acknowledgements

We are grateful to Daniel Valesin for the careful reading of the first version of this article and for their very helpful comments. The work of the two first authors was supported by a grant from the Swiss NSF and the NCCR SwissMap also funded by the Swiss NSF. The project was initiated during a stay of the third author to the Université de Genève, and the authors are grateful to the institution for making such a stay possible. AT was supported by CNPq grants 306348/2012-8 and 478577/2012-5 and by FAPERJ grant 202.231/2015.

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Authors and Affiliations

  1. Institut des Hautes Études Scientifiques, Le Bois-Marie 35, route de Chartres, 91440, Bures-sur-Yvette, France

    H. Duminil-Copin

  2. Département de mathématiques, Université de Genéve, 2-4 rue du lièvre, 64, 1211, Geneva 4, Switzerland

    H. Duminil-Copin & V. Tassion

  3. Instituto Nacional de Matemática Pura e Aplicada - IMPA, Estrada Dona Castorina 110, Rio de Janeiro, RJ, 22460-320, Brazil

    A. Teixeira

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  1. H. Duminil-Copin
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  2. V. Tassion
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  3. A. Teixeira
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Correspondence to A. Teixeira.

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Duminil-Copin, H., Tassion, V. & Teixeira, A. The box-crossing property for critical two-dimensional oriented percolation. Probab. Theory Relat. Fields 171, 685–708 (2018). https://doi.org/10.1007/s00440-017-0790-0

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  • DOI: https://doi.org/10.1007/s00440-017-0790-0

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