Extreme paths in oriented two-dimensional percolation

Abstract

International audienceA useful result about leftmost and rightmost paths in two dimensional bond percolation is proved. This result was introduced without proof in \cite{G} 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

The randomly fluctuating hyperrectangles are spatially monotone

“…In regard to the basic onedimensional contact process, which is the continuous-time analog of two-dimensional oriented percolation, Gray [G91] introduced the spatial monotonicity of its occupied site probabilities, among other intriguing properties regarding them. The detailed and elaborate proof of this notable result is given by Andjel and Gray [AG16] and by Andjel and Sued [AS08]. Whereas the corresponding property holds for undirected percolation on integer lattices is in general an open question.…”

“…and elaborate proof of this notable result is given by Andjel and Gray [AG16] and by Andjel and Sued [AS08]. Whereas the corresponding property holds for undirected percolation on integer lattices is in general an open question.…”