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