Baire categorical aspects of first passage percolation

TL;DR

This paper explores the topological structure of first passage percolation, revealing that some probabilistic properties have residual counterparts, while others become highly chaotic when viewed through a topological lens.

Contribution

It introduces a topological perspective to first passage percolation, analyzing residuality and chaos in classical probabilistic events without relying on probability measures.

Findings

Existence of geodesics has residual counterparts.

Limit shape and time constants become highly chaotic.

Certain probabilistic properties persist in a topological setting.

Abstract

In the previous decades, the theory of first passage percolation became a highly important area of probability theory. In this work, we will observe what can be said about the corresponding structure if we forget about the probability measure defined on the product space of edges and simply consider topology in the terms of residuality. We focus on interesting questions arising in the probabilistic setup that make sense in this setting, too. We will see that certain classical almost sure events, as the existence of geodesics have residual counterparts, while the notion of the limit shape or time constants gets as chaotic as possible.