The Bethe Lattice: A Tree-Like Structure

Interactive Bethe Lattice

The Bethe lattice is a special tree-like structure where each site has exactly z neighbors. Unlike regular lattices, it contains no loops - there is exactly one path between any two sites.

Coordination number (z): 3

Generations: 4

Structure Properties

• Each site has z neighbors (except boundary)
• No loops or cycles in the structure
• Unique path between any two sites
• z = 2 reduces to 1D chain

Growth Properties

• Sites at level l: $z(z-1)^{l-1}$
• Total sites to level l: $1 + \sum_{k=1}^l z(z-1)^{k-1}$
• Exponential growth with level
• No natural Euclidean distance

Hover over nodes to highlight branches and sub-branches. Notice how the structure grows exponentially with each generation, and how each site (except the boundary) maintains exactly z neighbors. The chemical distance between two sites is defined as the length of the unique path connecting them.