general hypercubic - has 2 times D neighbours in dimension D
Cayley tree - also known as the Bethe lattice because the Bethe
approximation is exact here. Each site except the surface sites have
z neighbours, z =3,4,...... It is like an infinite
dimension because in d dimensions the volume of a ``circle'' with
radius r is proportional to rd
while its surface is proportional to rd-1 .
As dimension tends to infinity the surface is proportional to the volume, as for the Cayley tree. A Cayley tree of z=3 is shown here.
amorphous/random - an ideal random lattice has points whose coordinates are randomly located. In physical systems there is usually some sort of short range order.
Universality - a simplistic definition which is adequate for us now is:
different models fall in the same universality class if
their critical exponents only depend on the geometric dimension
and not on the lattice details. (Critical temperatures will always be lattice dependent.)
Potts models - this a general family of models which includes both
Ising models and percolation as special cases. The formal defintion can be seen here.
The Ising model is the two state ( q=2) Potts model (think of spin up as one state and spin down as the other)
and percolation is formally described to be the q approaches 1 limit.
This is true in the formal sense that if one writes out expressions for the
model in series expansions, taking the limit as q approaches 1 gives percolation results.
Perhaps the easisst way to understand this is to define the 3 state model.Here there are three possible states or say, colors red green and blue.
Although Potts models do not have a solution like the Onsager one,
in two dimensions their critical points, exponents and transition
order are known exactly. (First order transition for q greater than 4.)
The critical points were calculated by Potts for
the square lattice using the idea of duality.
Duality is a concept from graph theory that means that each bond of the dual
lattice intersects a bond of the original lattice once and once only
and vice -versa. The square lattice is self-dual and the dual of the
triangular is the honeycomb lattice.