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

TL;DR

This paper proves a Russo-Seymour-Welsh type result for critical oriented percolation on a square lattice, revealing polynomial decay of connection probabilities and sub-linear fluctuations of critical clusters, extending to the contact process.

Contribution

It introduces a new RSW-type result for critical oriented percolation and characterizes the polynomial decay and fluctuation behavior of critical clusters, contrasting with non-oriented cases.

Findings

Connection probability decays polynomially with distance n

Critical cluster width scales between n^{2/5} and n^{1- ext{epsilon}}

Results extend to the graphical representation of the contact process

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 the origin conditioned to survive to distance $n$ has a typical width $w_n$ satisfying $\epsilon n^{2/5} < w_n < n^{1-\epsilon}$ for some $\epsilon > 0$. 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.