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The box-crossing property for critical two-dimensional oriented percolation. (English) Zbl 1428.60138
Summary: 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:
\(\bullet\)
We establish that the probability that the origin is connected to distance \(n\) decays polynomially fast in \(n\).
\(\bullet\)
We prove that the critical cluster of 0 conditioned to survive to distance \(n\) has a typical width \(w_n\) satisfying \(\varepsilon n^{2/5}\leq w_n\leq n^{1-\varepsilon}\) 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.

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B43 Percolation
82C43 Time-dependent percolation in statistical mechanics
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