To rigorously define the graph (assuming the percolation model is established in -dimensional space), we consider all points in as the vertices of the graph. These points can also be viewed as vectors, and we will not distinguish between these interpretations. Here, represents the -th component of .
To formally describe the edges in the graph, we define the -distance between vertices and as:
We also define as the -norm of the vector , that is:
In addition, we define as the -norm of , that is:
The edges of the graph in the cubic lattice are defined as all pairs of vertices satisfying . The set of all such edges is denoted by . In particular, any pair of vertices with are called adjacent vertices. Thus, the cubic lattice is rigorously defined as .
Next, we incorporate probability into the graph model. The definition of probability essentially involves the definition of a probability space. Since randomness in the simplest model arises solely from the open or closed states of edges, we define sample points as follows: A sample point is defined as an infinite-dimensional vector (with components), where each component takes a value of or . A value of indicates that the corresponding edge in the cubic lattice is open, while a value of indicates that the edge is closed. (This can be understood as labeling all the edges in the cubic lattice, which are countable, and recording whether each edge is open or closed in . In other words, each corresponds to a specific realization of the percolation model, aligning perfectly with the definition of a sample point.)
The sample point is also called an edge configuration, as it captures the state of every edge in the cubic lattice. For a specific edge , denotes the value of the component corresponding to , with if and only if the edge is open, and if and only if is closed.