1.2. JULIA SETS RELATED TO POTTS MODELS 7
functions are Julia sets of a family of rational maps, which are renormalization
transformations with respect to these physical models.
The hierarchical sequence of a generalized diamond lattice depends on two
natural parameters m ≥ 2 and n ≥ 2. The lattice Γ1 is just two ”outer” sites
related by a bond. In order to obtain Γ2, we connect two outer sites by m branches
of n bonds. Furthermore, in order to obtain Γj+1, we replace each bond in Γj by
the lattice Γ2 (j = 1, 2, · · · ) (see Fig 1.1 for m = 2 and n = 3).
Fig 1.1. Γ1, Γ2 and Γ3 for m = 2 and n = 3.
The above generalized diamond lattice is obviously a generalization of the stan-
dard diamond lattice (m = n = 2) and the diamond-like lattice (m ∈ N and n = 2).
The Potts model was initially defined for an integer λ as a generalization of the
Ising model (λ = 2) to more than two components (see [HU]). Later on, it was
shown that the Potts model for non-integer values λ may describe properties of a
number of physical systems such as dilute spin glasses, gelation and vulcanization
of branched polymers (0 λ 1). Also it was shown that bond and site percola-
tion problems could be formulated in terms of Potts models with pair and multisite
interactions in the λ = 1 limit (see [GA], [GU], [LI], [MO1], [MO2]).
At each site of the generalized diamond lattice we put a Potts spin which
can have λ different states, we then get a λ-state Potts model on the generalized
diamond hierarchical lattice. The Hamiltonian of the λ-state Potts model is
H = −J
δ(σi,σj), σi = 1, 2, · · · , λ
where δ is the Kronecker delta, and the sum is over nearest neighbors, and J is the
exchange interaction constant. The partition function is then given by
(1.1) Z =
where K = J
, here k is the Boltzmann constant and T is the temperature. This
describes ferromagnetic interaction for J 0, and antiferromagnetic interaction for
J 0 (see [HU]).
For convenience, we use the variable z = eK . In the following, we shall show
that the limiting set of the zeros of the partition function is the Julia set J(Umnλ)
of the rational map
(1.2) w = Umnλ(z) =
(z + λ −
+ (λ − 1)(z −
(z + λ − 1)n − (z − 1)n