# A phase transformation for the number of optimal paths in first passage percolation

## TL;DR

This paper investigates the behavior of the number of optimal paths in first passage percolation on a square lattice, revealing a phase transition at critical and sub-critical regimes.

## Contribution

It introduces a phase transition framework for the number of optimal paths in first passage percolation, highlighting new critical phenomena.

## Key findings

- Existence of a phase transition in the number of optimal paths
- Different behaviors in sub-critical and critical regimes
- Insights into the structure of optimal paths at large distances

## Abstract

We consider the first passage percolation model on the square lattice with an edge weight distribution F. In this paper, we consider the number of optimal paths for two points separated by a long distance. We show that there is a phase transition in the sub-criticality and the criticality.