Symmetry is a very important concept for physicists. A symmetry of an object or physical system is some sort of transformation which leaves it unchanged (or only changes it in a "nice" way). A sphere has rotational symmetry, as any rotation of a sphere about its center leaves it the same. A square lattice has translational symmetry, meaning that you can slide the lattice to a new position where every vertex and edge lines up with a vertex or edge in the original position. Symmetries can also be more abstract, like the transformations we explored in the previous section. Duality can be considered a type of symmetry that two-dimensional lattices possess.
The analysis of most real physical systems is somewhat different from our lattice problem. Energy dictates the distribution of possible states of the system, instead of our arbitrary open bond probability. Symmetry contrains the way that a system's energy can depend on its state - if a system is unchanged by a transformation, its energy must be, too. In a magnet, for example, the energy shouldn't change if we rotate all magnetic moments by the same amount (unless we are investigating the interaction with an external field). These constraints on the energy lead to very strong relationships between systems with the same symmetries, such as the equality of certain critical exponents which control how quantities like the energy of the system and its response to external fields change near a critical point. Our percolation system has a critical exponent: the percolation probability at a probability \(p\) is proportional to \((p-p_c)^\beta\) just above the critical probability \(p_c\) for some exponent \(\beta\). Unfortunately, this exponent is fairly challenging to compute for a two-dimensional lattice, and simulations don't give good approximations until they are extremely large.
There is a simpler universal property for lattice percolation, which you might have noticed already. If we multiply the critical probability for a lattice by the number of neighbors of each cell, we get roughly 2 in each of the three lattice we considered!
| Lattice | Diamond | Simple Cubic | Body-Centered Cubic | Face-Centered Cubic | Hexagonal Close-Packing |
|---|---|---|---|---|---|
| #Neighbors | 4 | 6 | 8 | 12 | 12 |
| pc | 0.388 | 0.247 | 0.178 | 0.119 | 0.124 |
| pc*#Neighbors | 1.55 | 1.48 | 1.42 | 1.43 | 1.49 |
The table above shows the same quantities for some three-dimensional lattices which are common in metals. Multiplying the critical probability by the number of neighbors seems to give 1.5 instead of 2. These results suggest that the percolation problem on lattices of different dimensions are universality classes, and thus that the nature of percolation close to a critical point depends on the dimension of the lattice rather than on its specific structure.