A useful result about leftmost and rightmost paths in two-dimensional bond percolation is proved. This result was introduced without proof in Gray (1991) in the context of the contact process in continuous time. As discussed here, it also holds for several related models, including the discrete-time contact process and two-dimensional site percolation. Among the consequences are a natural monotonicity in the probability of percolation between different sites and a somewhat counter-intuitive correlation inequality.
Bibliographical noteFunding Information:
We thank the anonymous referee for a careful reading and helpful comments. This paper was written while the first author was visiting IMPA and was supported first by CAPES (grant number 1331/11-6) and then by CNPq (grant number 300722/2013-3).
- Extreme paths
- Oriented percolation