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Bond percolation is a mathematical model used to study the connectivity of a network or lattice. In this model, each bond (or edge) between nodes (or vertices) is independently occupied with a certain probability p. The goal is to determine the critical probability pc at which a giant connected cluster of occupied bonds emerges, spanning the entire lattice. This phase transition is known as the percolation threshold.

Bond percolation can be studied on various types of lattices, such as square, triangular, or hexagonal lattices, as well as on more complex networks. The critical probability pc depends on the lattice structure and dimensionality. For example, on a square lattice, pc is approximately 0.5, while on a triangular lattice, it is about 0.6527.

The behavior of bond percolation can be analyzed using various techniques, including numerical simulations, analytical methods, and renormalization group theory. One of the key results is the scaling behavior of the system near the percolation threshold. The size of the largest cluster, the number of clusters, and other properties exhibit power-law scaling with exponents that depend on the dimensionality and lattice structure.

Bond percolation has applications in various fields, including physics, computer science, and materials science. It can be used to model the spread of diseases, the flow of fluids through porous media, and the behavior of complex networks. Understanding the percolation threshold and the scaling behavior of bond percolation can provide insights into the connectivity and robustness of these systems.