[Paper Review] Extreme paths in oriented 2D Percolation

Enrique D. Andjel, Lawrence Gray|arXiv (Cornell University)|Nov 4, 2014

Stochastic processes and statistical mechanics2 references3 citations

TL;DR

This paper establishes a rigorous foundation for leftmost and rightmost paths in two-dimensional oriented bond percolation, proving a key structural result previously stated without proof. The result enables monotonicity in percolation probabilities and reveals a counter-intuitive correlation inequality, extending to models like the discrete-time contact process and site percolation.

ABSTRACT

A 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.

Motivation & Objective

• To rigorously prove the existence and properties of leftmost and rightmost paths in two-dimensional oriented bond percolation.
• To extend the validity of these path properties to related models, including the discrete-time contact process and two-dimensional site percolation.
• To establish a monotonicity property in percolation probabilities between different sites based on path structure.
• To derive a non-trivial correlation inequality that defies intuitive expectations, arising from the extreme path structure.

Proposed method

• Analyzes oriented bond percolation on a 2D lattice, focusing on directed paths from the origin.
• Defines leftmost and rightmost paths as extremal paths in the set of all directed paths from the origin.
• Uses path comparison techniques to show that these extremal paths are almost surely unique and well-defined.
• Applies the path structure to derive stochastic monotonicity in percolation probabilities across spatial locations.
• Derives a correlation inequality by comparing the influence of different path extremities on percolation events.
• Extends the results to other models via coupling and duality arguments, including the discrete-time contact process and site percolation.

Experimental results

Research questions

• RQ1 What structural properties govern the behavior of leftmost and rightmost directed paths in 2D oriented bond percolation?
• RQ2 How do these extremal paths lead to monotonicity in the probability of percolation between different lattice sites?
• RQ3 What non-trivial correlation inequalities emerge from the geometry of extreme paths in percolation?
• RQ4 To what extent do these results generalize to other stochastic processes like the discrete-time contact process?
• RQ5 How does the orientation of bonds affect the existence and uniqueness of extremal paths in the percolation regime?

Key findings

• The leftmost and rightmost paths in 2D oriented bond percolation are almost surely unique and well-defined under the model's dynamics.
• Percolation probabilities exhibit a natural monotonicity: if site y is to the right of site x, the probability of percolation to y is at least as high as to x.
• A counter-intuitive correlation inequality is derived, showing that the occurrence of a leftmost path increases the likelihood of a rightmost path under certain conditions.
• The structural results on extreme paths extend beyond bond percolation to include the discrete-time contact process and two-dimensional site percolation.
• The proof technique relies on path dominance and comparison arguments, establishing a robust framework for analyzing extremal paths.
• The findings provide a foundational tool for studying path dominance and spatial dependence in oriented percolation and related interacting particle systems.