Lower large deviations for the maximal flow through tilted cylinders in two-dimensional first passage percolation

TL;DR

This paper establishes the lower large deviation principles for the maximal flow in two-dimensional first passage percolation, showing deviations are of surface order and extending previous results to tilted rectangles.

Contribution

It extends and improves prior large deviation results by proving the lower large deviations are of surface order for tilted rectangles in 2D first passage percolation.

Findings

Lower large deviations are of surface order.

Established the large deviation principle from below.

Extended results to tilted rectangles, not just aligned boxes.

Abstract

Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. A law of large numbers is known for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. We prove that the lower large deviations are of surface order, and we prove the corresponding large deviation principle from below. This extends and improves previous large deviations results of Grimmett and Kesten (1984) obtained for boxes of particular orientation.