Chemical distance for the half-orthant model

Abstract

The half-orthant model is a partially oriented model of a random medium involving a parameter $p\in [0,1]$, for which there is a critical value $p_c(d)$ (depending on the dimension $d$) below which every point is reachable from the origin. We prove a limit theorem for the graph-distance (or "chemical distance") for this model when $p<p_c(2)$, and also when $1-p$ is larger than the critical parameter for site percolation in $\mathbb{Z}^d$. The proof involves an application of the subadditive ergodic theorem. Novel arguments herein include the method of proving that the expected number of steps to reach any given point is finite, as well as an argument that is used to show that the shape is "non-trivial" in certain directions.

Figures (4)

• Figure 1: An example of a finite piece of the environment for the half-orthant model, with the shortest path from the origin (centre) to $(-1,1)$ highlighted.
• Figure 2: Top row: Simulation of the set of points whose distance from the origin within the random environment is exactly $n=4000$, for $p=0.25, p=0.5,p=0.75$ (left to right) respectively. Bottom row: Simulations of these set for different values of $n$, for $p=0.25, p=0.5,p=0.75$. Note the apparent flat and curved regions in these shapes.
• Figure 3: An illustration of the region $\mathcal{S}_p^{\textrm{good}}$ in 3 dimensions. The $\ell_1$ ball in 3 dimensions appears on the left, with an example of $\mathcal{S}_p^{\textrm{good}}$ depicted on the right.
• Figure 4: Simulation of the set of points reachable in $n$ steps for the orthant model with $p=1/2$ (left) and $p=3/4$ (right). White patches surrounded by colour are not reachable from the origin.

Theorems & Definitions (27)

• Theorem 1
• Corollary 1
• Theorem 2
• Remark 1
• Remark 2: Backward connections
• Theorem 3: Kingman, Liggett
• Lemma 1
• proof
• Lemma 2
• Lemma 3