Classification of the limit shape for 1+1-dimensional FPP

TL;DR

This paper studies a simplified 1+1-dimensional first passage percolation model, revealing conditions for flat edges in the limit shape based on the distribution of edge weights, and providing bounds on the time constant derivatives.

Contribution

It introduces a simplified planar FPP model with deterministic vertical weights and characterizes when the limit shape has a flat edge based on the horizontal edge distribution.

Findings

Flat edge in the limit shape occurs iff the horizontal edge distribution has an atom at its infimum.

Provides bounds on the upper and lower derivatives of the time constant.

Establishes a precise condition linking edge weight distribution to the shape of the limit.

Abstract

We introduce a simplified model of planar first passage percolation where weights along vertical edges are deterministic. We show that the limit shape has a flat edge in the vertical direction if and only if the random distribution of the horizontal edges has an atom at the infimum of its support. Furthermore, we present bounds on the upper and lower derivative of the time constant.