The sum is convergent since | tr M
n
| ≤ kMk
n ∞
. Furthermore, the assumed bound |u| kMk
−1 ∞
also guarantees the absolute convergence of the sums obtained by splitting: tr M
n
= X
e
1
,...,e
n
∈ b E
M
e
n
,e
1
×
n−1
Y
j=1
M
e
j
,e
j+1
2.4 =
X
e∈ b E
X
b γ∋e
|b γ|=n
χ
M
bγ =
∞
X
k=1
X
p
δ
k|p|,n
|p| χ
M
p
k
,
where the last equation is obtained by representing each loop bγ as the suitable power k of a
primitive oriented loop and p ranges over equivalence classes of such, modulo cyclic permutation.
Thus
∞
X
n=1
u
n
n tr M
n
=
∞
X
k=1
X
p
1 k
h u
|p|
χ
M
p i
k
= − X
p
ln 1 − u
|p|
χ
M
p .
2.5 Combined with 2.3 this yields 2.2.
The meromorphic function ζ
M
defined in 2.2 is related to Ihara’s graph zeta function, whose usual definition is in terms of site-indexed paths and
V × V matrices. In that case the customary non-backtracking restriction on
p needs to be added explicitly. Also the statement and proof of the corresponding result are a bit more involved. Further background on this topic, other extensions,
and graph theoretic applications, can be found in, e.g., [35, 28]. In view of the simplicity of the argument, it may be worth stressing that the limitations placed on
the loops in 2.2 leave the possibility of self intersection, multiple crossing of edges, and arbitrary repetitions of sub-loops. The product in 2.2 is over an infinite collection of factors, whose series
expansions yield terms with arbitrary powers of u. However on the left side is a polynomial in u.
Thus contained in this zeta function relation is an infinite collection of combinatorial cancellations.
3. The Ising model
3.1. The model’s basic variables
The Ising model is a system of ±1 valued spin variables σ
x
attached to the vertices V of a graph G with the energy function
Hσ = − X
{x,y}∈E
J
x,y
σ
x
σ
y
− h X
x∈V
σ
x
3.1 given in terms of edge couplings
J
x,y
. The probability distribution representing thermal equilib- rium is the Gibbs measure, referred to as the Gibbs state,
Probσ = e
−βHσ
Z
G
β, h with
Z
G
β, h = P
σ
e
−βHσ
3.2
3
for which the expectation value functional will be denoted by h·i
G β,h
some of whose indices will occasionally be omitted. The normalizing factor is the partition function
Z
G
β, h. Through it, one computes the thermodynamic pressure:
ψβ, h = 1
|V| log Z
G
β, h . 3.3
Of particular interest is the infinite volume limit |V| → ∞ for the pressure, for which singularities
may develop corresponding to phase transitions. Of further interest are the infinite volume limits of the Gibbs equilibrium states. These can be viewed as tangents of the pressure functional [31],
and unlike ψβ, h their limits may depend on the boundary conditions.
For most of the analysis which follows, the model’s coupling constants need not be constant, or ferromagnetic. The first of these restriction will be invoked only in extracting Onsager’s formula
for the free energy from the Kac-Ward determinantal expression for the case G = Z
2
and J
x,y
= J for all
{x, y} ∈ EZ
2
.
3.2. A graphical high temperature representation
In discussing the model’s partition function it is convenient to split off a trivial factor and denote: e
Z
G
β, h := Z
G
β, h
2
|V|
Y
{x,y}⊂E
coshβJ
x,y
.
3.4 For any collection of edges
Γ ⊂ E, we denote by ∂Γ the collection of vertices of the graph which are oddly covered by
Γ. Edge collections with ∂Γ = ∅ correspond to so called even subgraphs of G. In these terms one has:
Lemma 3.1. For any finite graph G the Ising partition function at vanishing field admits the fol-
lowing expansion into even subgraphs: e
Z
G
β, h = 0 = X
Γ⊂E : ∂Γ=∅
wΓ 3.5
with wΓ :=
Q
{x,y}∈Γ
W
{x,y}
at the weights W
{x,y}
= tanhβJ
x,y
. 3.6
The corresponding spin correlation functions of any even number of vertices A ⊂ V are represented
by h
Y
x∈A
σ
x
i
β,h=0
e Z
G
β, h = 0 = X
Γ : ∂Γ=A
wΓ . 3.7
This well known representation is obtained by writing e
βJ
x,y
σ
x
σ
y
= coshβJ
x,y
1 + σ
x
σ
y
tanhβJ
x,y
,
4
expanding Y
{x,y}⊂E
1 + σ
x
σ
y
tanh βJ
x,y
= X
Γ
Y
{x,y}∈Γ
[σ
x
σ
y
tanhβJ
x,y
] , and then summing the resulting expression over the spin configurations
σ. Equation 3.5 results from the relation
P
σ
Q
{x,y}∈Γ
σ
x
σ
y
= 2
|V|
1[∂Γ = ∅] with 1[. . . ] denoting the indicator function. The second identity 3.7 follows by similar arguments; see also [17, 11].
4. Onsager’s solution and the Kac-Ward formula