2. Graph zeta functions
2.1. Paths on graphs with transition-based weights
For a graph, described in terms of its vertex and edge set as G = V, E, the set of its oriented
edges will be denote here by b
E ≡ b EG = E × {+1, −1}. These will be used to describe paths and
loops on G. The presentation of paths in this form, rather than through sequences of visited sites,
simplifies the formulation of weights which depend multiplicatively on both the edges visited and turns taken.
Definition 2.1. Paths and loops – basic terminology and notation 1. An oriented edge
e ≡ x
1
, x
2
is said to lead into e
′
≡ y
1
, y
2
if y
1
= x
2
. The relation is denoted
e ⊲ e
′
. The reversal of e is denoted e.
2. A path on G is a finite sequence bγ = e
, e
1
, . . . , e
n
of oriented edges sequentially lead- ing into each other. The number of such steps,
n, is denoted by |bγ|. A path is said to be backtracking if
e
j+1
= e
j
for some j.
3. A loop is an equivalence class, modulo cyclic reparametrization, of non-backtracking paths e
, e
1
, . . . , e
n
with e
n
= e . A loop is called primitive if it cannot be presented as some
k-fold repetition of a shorter one. 4. A b
E × b E matrix M is non-backtracking if M
e,e
= 0 for all e ∈ b E. For a b
E × b E matrix M
and a path or loop bγ = e
, e
1
, . . . , e
n
, we denote: χ
M
bγ :=
n
Y
j=1
M
e
j−1
,e
j
. 2.1
which for loops is invariant under cyclic reparametrizations. The following general results allows a natural introduction of the Bowen-Lanford zeta function
ζ
M
[5]. Theorem 2.2. Let
G be a finite graph and M a non-backtracking b E × b
E matrix. Then for all u ∈ C such that
|u|
−1
kMk
∞
:= max
e∈ b E
P
e
′
∈ b E
|M
e,e
′
| : det1 − uM =
Y
p
h 1 − u
|p|
χ
M
p i
=: ζ
M
u
−1
2.2 where the product ranges over the primitive oriented loops, each equivalence class contributing a
single factor. Proof.
For u small enough ln det1 − uM is analytic in u and satisfies:
ln det1 − uM = tr ln1 − uM = −
∞
X
n=1
u
n
n tr M
n
. 2.3
2
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
Kategori