Law of large numbers for the maximal flow through tilted cylinders in two-dimensional first passage percolation

TL;DR

This paper establishes a law of large numbers for the maximal flow through large tilted rectangles in 2D first passage percolation, extending previous results to more general orientations.

Contribution

It generalizes the law of large numbers for maximal flow to tilted rectangles, depending on the aspect ratio, broadening prior orientation-specific results.

Findings

Law of large numbers for maximal flow in tilted rectangles

Dependence of the limit on the aspect ratio of the rectangle

Extension of previous orientation-specific results

Abstract

Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. We prove a law of large numbers for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. The value of the limit depends on the asymptotic behaviour of the ratio of the height of the cylinder over the length of its basis. This law of large numbers extends the law of large numbers obtained by Grimmett and Kesten (1984) for rectangles of particular orientation.