Anomalous behavior of replicator dynamics for the prisoner's dilemma on diluted lattices

In diluted lattices, cooperation is often enhanced at specific densities, particularly near the percolation threshold for stochastic updating rules. However, the replicator rule, despite being probabilistic, does not follow this trend. We find that this anomalous behavior is caused by structures formed by holes and defectors, which prevent some agents from experiencing fluctuations, thereby restricting the free flow of information across the network. As a result, the system becomes trapped in a frozen state, though this can be disrupted by introducing perturbations. Finally, we provide a more quantitative analysis of the relationship between the percolation threshold and cooperation, tracking its development within clusters of varying sizes and demonstrating how the percolation threshold shapes the fundamental structures of the lattice.