# Distribution of the Number of Paths in Two-Dimensional Directed Percolation

## TL;DR

This paper numerically analyzes the number of directed percolating paths in two-dimensional systems and explores their distribution across different phases.

## Contribution

The study provides detailed numerical analysis of path distributions in percolation using large-deviation methods to reach extremely low probabilities.

## Key findings

- The average entropy 〈S〉 was studied as a function of occupation density p.
- The distribution P(S) was computed down to probabilities as low as 10−300.
- Structural analysis of realizations was performed for various values of S and p.

## Abstract

In a percolating system, there are typically exponentially many spanning paths. Here, we study numerically, for a two-dimensional L×L diluted system, restricted to percolating realizations, the number N of directed percolating paths. First, we study the average entropy 〈S〉=〈logN〉 as a function of the occupation density p and compare with mathematical results from the literature. Furthermore, we investigate the distribution P(S). By using large-deviation approaches, we are able to obtain P(S) down to the very low-probability tail reaching probabilities as small as 10−300. We consider the percolating phase, the (typically) non-percolating phase, and the critical point. Finally, we also analyze the structure of the realizations for some values of S and p.