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

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.