# Graph Zeta Functions
and
Wilson Loops in Kazakov-Migdal Model

So Matsuura<EMAIL_ADDRESS>Hiyoshi Departments of Physics, and Research and
Education Center for Natural Sciences, Keio University, 4-1-1 Hiyoshi,
Yokohama, Kanagawa 223-8521, Japan Kazutoshi Ohta<EMAIL_ADDRESS>Institute of Physics, Meiji Gakuin University, Yokohama, Kanagawa 244-8539,
Japan

Abstract

In this paper, we consider an extended Kazakov-Migdal model defined on an
arbitrary graph. The partition function of the model, which is expressed as
the summation of all Wilson loops on the graph, turns out to be represented by
the Bartholdi zeta function weighted by unitary matrices on the edges of the
graph. The partition function on the cycle graph at finite $N$ is expressed by
the generating function of the generalized Catalan numbers. The partition
function on an arbitrary graph can be exactly evaluated at large $N$ which is
expressed as an infinite product of a kind of deformed Ihara zeta function.
The non-zero area Wilson loops do not contribute to the leading part of the
$1/N$-expansion of the free energy but to the next leading. The semi-circle
distribution of the eigenvalues of the scalar fields is still an exact
solution of the model at large $N$ on an arbitrary regular graph, but it
reflects only zero-area Wilson loops.

##  1 Introduction

Wilson loops [1] are key objects to understand the non-perturbative dynamics
of non-Abelian gauge theories. Since the Wilson loop (and the Polyakov loop)
is related to the potential between the (infinitely heavy) quark and anti-
quark, it is an important observable for approaching the quark confinement
problem, which is one of the main subjects of the non-Abelian gauge theory.

The Wilson loops are the most general gauge invariant operators in non-Abelian
gauge theory. In fact, the gauge theory can even be reformulated in terms of
the Wilson loops [2, 3]. It is therefore not a coincidence that the simplest
action of the lattice gauge theory is given by the plaquette variable, that
is, the smallest Wilson loop on the lattice. When the Wilson loop on the
lattice is expanded in terms of the lattice spacing, each term is expressed as
a product of the field strengths of the gauge fields. While the leading term
of the expansion of the plaquette action gives the gauge kinetic term
$\operatorname{Tr}F^{2}$ in the continuous theory, the expansion of larger
Wilson loops begins with higher-derivative operators. Thus, if we construct a
general gauge invariant action on the lattice by adding up several Wilson
loops, the irrelevant higher-derivative terms do not contribute in the
continuum limit in general and it is considered to describe the non-Abelian
gauge theories on the continuous space-time universally.

The so-called Kazakov-Migdal (KM) model [4] is a lattice gauge theory defined
on the $D$-dimensional lattice with the action,

$S_{\rm
KM}=N\sum_{x}\operatorname{Tr}\left\\{\frac{m^{2}}{2}\Phi^{2}(x)-\sum_{\mu=1}^{D}\Phi(x)U_{\mu}(x)\Phi(x+\mu)U_{\mu}^{\dagger}(x)\right\\}\,,$
(1.1)

where $U_{\mu}(x)$ is a unitary variable living on the link extending from the
site (vertex) $x$ to the direction of $\mu$ and $\Phi(x)$ is a scalar field in
the adjoint representation of $SU(N)$ living on $x$. Remarkably, the effective
action obtained by integrating out the scalar fields of this model is the sum
of all possible Wilson loops. Furthermore, this model can be solved exactly at
large $N$ thanks to its simple structure [5, 6]. Hence, this model first
attracted attention as a theory that can be solved at large $N$ and is
expected to induce QCD (more precisely, quantum “gluodynamics” without quarks)
[7, 8, 9, 10, 11, 12, 13].

However, it has become clear that the KM model does not actually induce QCD
[14, 15, 16]. The main difficulty is that this model has an extra
$\mathbb{Z}_{N}$ gauge symmetry111 For the gauge group is $U(N)$, it becomes
an $U(1)$ center of the gauge symmetry. ;
$U_{\mu}(x)\to\omega_{\mu}(x)U_{\mu}(x)$ so that $\omega_{\mu}(x)^{N}=1$,
which is not provided by Wilson’s lattice gauge theory [14]. Due to this
symmetry, a natural operator with finite expected value is not a single Wilson
loop $\langle\operatorname{Tr}W_{C}(U)\rangle$, which trivially vanishes, but
the square of its absolute value
$\langle|\operatorname{Tr}W_{C}(U)|^{2}\rangle$. This is not the situation
realized by the usual QCD. Despite various efforts to circumvent this
difficulty, there was no sign of QCD being realized in the continuous limit of
modified KM models [17, 18, 19, 20, 21].

Even if the KM does not induce QCD, however, this model is still an
interesting matrix model in its own right [22, 23, 24, 25, 26, 27]. In
particular, it is still true that the KM model counts all possible Wilson
loops on the lattice in the form of the square of the absolute value. In our
previous paper [28], we proposed a generalized KM model which is defined on an
arbitrary graph, which is called the graph KM (gKM) model. We showed that, by
tuning the parameters of the model appropriately, the partition function is
expressed as the integral of the Ihara zeta function [29, 30, 31] weighted by
unitary matrices on the edges of the graph. The Ihara zeta function is the
simplest of the graph zeta functions, which counts all reduced cycles on the
graph. Correspondingly, we can explicitly write down the effective action of
this parameter-tuned model as a theory that adds up all non-zero area Wilson
loops on a given graph. In particular, we can evaluate the partition function
exactly in the large $N$ limit thanks to the large $N$ factorization of the
Wilson loops and properties of the Ihara zeta function, which becomes an
infinite product of the (normal) Ihara zeta function in general. This result
suggests that the graph zeta function will be useful for the analysis of
lattice gauge theory and that the gKM model will have important implications
for the knowledge of the zeta function at the same time.

In the present paper, we relax the condition imposed in [28] and consider the
gKM model with two parameter degrees of freedom. As mentioned above, while the
Ihara zeta function counts only cycles without bumps (backtrackings and/or
tails), general cycles on a graph typically have bumps. A graph zeta function
that has been extended to count general cycles with bumps is known as the
Bartholdi zeta function [32] (see also [33]), which has two parameters that
count the length of the cycles and the number of bumps. The parameters of the
gKM model considered in this paper can be related to these parameters. We show
that the partition function of the gKM model in this general case is
represented by the integral of the Bartholdi zeta function with unitary matrix
weights on the edges.

The organization of this paper is as follows: In Sect. 2, we introduce the
terminology of the graph theory and review the basic properties of the Ihara
and Bartholdi zeta functions. We also propose a matrix weighted Bartholdi zeta
function. In Sect. 3, we introduce a generalized KM model on an arbitrary
graph (gKM model). We show that the partition function of the gKM model is
described by the integral of the weighted Bartholdi zeta function by unitary
matrices on the edges. We explicitly write down the two representations of the
partition function that arise by changing the order of integration over the
two kinds of matrices; hermitian matrices on the vertices and the unitary
matrices on the edges. We also see the relation to the original KM model. In
Sect. 4, we exactly evaluate the partition function of the gKM model on the
cycle graph at finite $N$. We see that the partition function is closely
related to the generating function of the generalized Catalan numbers. In
Sect. 5, we exactly evaluate the partition function of the model on an
arbitrary graph at large $N$. The formulas developed in the previous paper
[28] play essential roles. We show that the leading of the $1/N$ expansion of
the free energy starts from the contribution of the zero area Wilson loops. We
discuss the relation to the conventional large $N$ analysis of the original KM
model. Sect. 6 is devoted to the conclusions and the discussions. In the
appendix, we give proofs of Amitsur’s theorem and an important identity used
in Sect. 2.

##  2 Graph zeta functions and their extensions

###  2.1 Ihara and Bartholdi zeta function

Let us begin with an explanation of graph theory terminology. Suppose $G$ is a
connected directed graph that has $n_{V}$ vertices and $n_{E}$ edges. Let us
denote the set of vertices and edges as $V$ and $E$, respectively. A directed
edge is written by a pair of vertices $e=\langle u,v\rangle$, where $u=s(e)$
and $v=t(e)$ the “source” and “target” of the edge arrow of $e$, respectively.
A reversed arrow edge for $e$ is called the inverse of $e$ and denoted by
$e^{-1}=\langle v,u\rangle$. We combine the set of the edges with their
inverses as

$E_{D}=\\{{\boldsymbol{e}}_{a}|a=1,\cdots,2n_{E}\\}\equiv\\{e_{1},\cdots,e_{n_{E}},e^{-1}_{1},\cdots,e^{-1}_{n_{E}}\\}\,.$
(2.1)

In other words, $E_{D}$ is the direct sum of the set of edges $E$ and their
inverses $E^{-1}$; $E_{D}=E\oplus E^{-1}$.

A path $P=({\boldsymbol{e}}_{1},\cdots,{\boldsymbol{e}}_{k})$
$({\boldsymbol{e}}_{a}\in E_{D})$ is a sequence of the edges which satisfies
$t({\boldsymbol{e}}_{a})=s({\boldsymbol{e}}_{a+1})$ $(a=1,\cdots,k-1)$, where
$k$ is called the length of the path $P$ which is expressed as $|P|$. If two
paths $P=({\boldsymbol{e}}_{1},\cdots,{\boldsymbol{e}}_{k})$ and
$P^{\prime}=({\boldsymbol{e}}^{\prime}_{1},\cdots,{\boldsymbol{e}}^{\prime}_{l})$
satisfy $t({\boldsymbol{e}}_{k})=s({\boldsymbol{e}}^{\prime}_{1})$, we can
construct a new path of length $k+l$ by connecting as
$PP^{\prime}\equiv({\boldsymbol{e}}_{1}\cdots,{\boldsymbol{e}}_{k},{\boldsymbol{e}}^{\prime}_{1},\cdots,{\boldsymbol{e}}^{\prime}_{l})$.
A backtracking of $P$ is a part of $P$ which satisfies
${\boldsymbol{e}}_{a+1}^{-1}={\boldsymbol{e}}_{a}$.

When a path $P=({\boldsymbol{e}}_{1},\cdots,{\boldsymbol{e}}_{k})$ satisfies
$s({\boldsymbol{e}}_{1})=t({\boldsymbol{e}}_{k})$, $P$ is called a cycle of
length $k$. A cycle $C$ is called primitive when $C$ satisfies $C\neq B^{r}$
$(r\geq 2)$ for any cycle $B$. A cycle
$C=({\boldsymbol{e}}_{1},\cdots,{\boldsymbol{e}}_{k})$ is called tailless when
${\boldsymbol{e}}_{k}^{-1}\neq{\boldsymbol{e}}_{1}$, which is equivalent to
that $C^{2}$ has no backtracking. Backtracking or tail is also called bump,
and the number of bumps of a cycle $C$ is denoted by $b(C)$.

Two cycles $C=({\boldsymbol{e}}_{1},\cdots,{\boldsymbol{e}}_{k})$ and
$C^{\prime}=({\boldsymbol{e}}^{\prime}_{1},\cdots,{\boldsymbol{e}}^{\prime}_{k})$
are called equivalent when
${\boldsymbol{e}}_{a}={\boldsymbol{e}}^{\prime}_{a+r}$ for some integer $r$.
We denote the equivalence class including a cycle $C$ as $[C]$. A cycle $C$ is
called reduced when $C$ has neither backtracking nor tail. We also denote the
set of representatives of the equivalence classes of all kinds of cycles
containing bumps by $[{\mathcal{P}}]$, and we denote the set of
representatives of the equivalence classes of reduced cycles by
$[{\mathcal{P}}_{R}]\subset[{\mathcal{P}}]$.

The Ihara zeta function [29, 30, 31] associated with a connected graph $G$ is
defined as

$\zeta_{G}(q)=\prod_{C\in[{\cal P}_{R}]}\frac{1}{1-q^{|C|}}\,.$ (2.2)

Noting that, if $C$ is a primitive cycle, $C^{-1}$ is also a primitive cycle
with the same length as $C$, a set of the equivalence classes of the primitive
cycles of length $\ell$ can be decomposed into
$\Pi_{\ell}^{+}\sqcup\Pi_{\ell}^{-}$, where $\Pi_{\ell}^{-}$ is the set of the
inverse of the elements in $\Pi_{\ell}^{+}$ such that
$\Pi_{\ell}^{-}\equiv\left\\{C^{-1}|C\in\Pi_{\ell}^{+}\right\\}$. Since these
sets $\Pi_{\ell}^{+}$ and $\Pi_{\ell}^{-}$ have the same number of elements by
definition, namely $|\Pi_{\ell}^{+}|=|\Pi_{\ell}^{-}|$, the Ihara zeta
function (2.2) can be rewritten as

$\displaystyle\zeta_{G}(q)=\prod_{\ell=1}^{\infty}\frac{1}{(1-q^{\ell})^{2|\Pi^{+}_{\ell}|}}\,.$
(2.3)

In the following, we call the element of $\Pi_{\ell}^{+}$ as the equivalence
class of chiral primitive cycles of length $\ell$ and write the set of the
equivalence classes of all chiral primitive cycles as

$\Pi^{+}\equiv\bigcup_{l=1}^{\infty}\Pi_{l}^{+}\,.$ (2.4)

We can further rewrite (2.3) by using the identity
$(1-x)^{-1}=\exp(\sum_{m=1}^{\infty}\frac{x^{m}}{m})$. Since any reduced cycle
is represented by a positive power of a primitive cycle $C$ and the
equivalence class $[C^{n}]$ $(n\in{\mathbb{N}})$ has $|C|$ different elements,
the Ihara zeta function (2.2) can be regarded as a generating function of the
number of reduced cycles;

$\displaystyle\zeta_{G}(q)=\exp\left(\sum_{k=1}^{\infty}\frac{N_{k}}{k}q^{k}\right)\,,$
(2.5)

where $N_{k}$ is the number of the reduced cycles of length $k$.

Although we have considered only reduced cycles so far, we can define a
function that counts up all cycles containing the bumps as

$\zeta_{G}(q,u)=\prod_{C\in[{\cal P}]}\frac{1}{1-q^{|C|}u^{b(C)}}\,,$ (2.6)

where the product is taken over the equivalence class of all the cycles
including bumps. This function is called the Bartholdi zeta function
associated with a connected graph $G$ [32]222See also [33] for a
generalization.. Note that, if we take $u=0$, the Bartholdi zeta function
reduces to the Ihara zeta function;

$\zeta_{G}(q,u=0)=\zeta_{G}(q)\,.$

It is remarkable that the Bartholdi zeta function (and thus also the Ihara
zeta function) is represented as the inverse of a polynomial, even though the
graph generally has infinitely many equivalence classes of cycles. The key is
the following theorem [34, 35] (see Appendix A for a proof):

Theorem (Amitsur). Let us consider “letters” $1,\cdots,k$ and call a sequence
of the letters a word. In particular, we call such a word that cannot be
written as a proper power of a shorter word “primitive”. We call two words
$w_{1}$ and $w_{2}$ equivalent when $w_{2}$ is obtained by a cyclic rotation
of the letters of $w_{1}$. We then denote the set of the representatives of
the equivalent primitive words $L$. For square matrices $X_{1},\cdots,X_{k}$,
we define $X_{w}\equiv X_{i_{1}}\cdots X_{i_{n}}$ corresponding to a word $w$.
Then the identity;

$\det\left(\boldsymbol{1}-X_{1}-\cdots-X_{k}\right)=\prod_{w\in
L}\det\left(\boldsymbol{1}-X_{w}\right)$

holds.

In particular, the following corollary is useful:

Corollary. Let ${X}$ be a square matrix of the block form,

${X}=\begin{pmatrix}X_{11}&\cdots&X_{1k}\\\ \vdots&\ddots&\vdots\\\
X_{k1}&\cdots&X_{kk}\end{pmatrix}\,,$

where $X_{ij}$’s are also square matrices of the same size $l$. For a word
$w=i_{1}\cdots i_{n}\in L$, we define

$X_{w}=X_{i_{1}i_{2}}X_{i_{2}i_{3}}\cdots X_{i_{n-1}i_{n}}X_{i_{n}i_{1}}.$

Then the identity,

$\det\left(\boldsymbol{1}_{kl}-X\right)=\prod_{w\in
L}\det\left(\boldsymbol{1}_{l}-X_{w}\right)\,,$

holds.

To rewrite the Bartholdi zeta function (2.6) (and the Ihara zeta function
(2.2)), we define $2n_{E}\times 2n_{E}$ matrices $W$ and $J$ whose elements
are defined by

$\displaystyle
W_{{\boldsymbol{e}}{\boldsymbol{e}}^{\prime}}=\begin{cases}1&{\rm if}\
t({\boldsymbol{e}})=s({\boldsymbol{e}}^{\prime})\ {\rm and}\
{\boldsymbol{e}}^{\prime-1}\neq{\boldsymbol{e}}\\\ 0&{\rm
others}\end{cases}\,,\quad
J_{{\boldsymbol{e}}{\boldsymbol{e}}^{\prime}}=\begin{cases}1&{\rm if}\
{\boldsymbol{e}}^{\prime-1}={\boldsymbol{e}}\\\ 0&{\rm
others}\end{cases}\,,\quad$ (2.7)

where ${\boldsymbol{e}},{\boldsymbol{e}}^{\prime}\in E_{D}$, which are called
the edge adjacency matrix and the bump matrix, respectively. Using these
matrices, we now consider a combination of the matrix $W$ and $J$ as $X\equiv
q(W+uJ)$ whose elements are given by

$X_{{\boldsymbol{e}}{\boldsymbol{e}}^{\prime}}=\begin{cases}qW_{{\boldsymbol{e}}{\boldsymbol{e}}^{\prime}}&{\boldsymbol{e}}^{\prime}\neq{\boldsymbol{e}}^{-1}\\\
quJ_{{\boldsymbol{e}}{\boldsymbol{e}}^{\prime}}&{\boldsymbol{e}}^{\prime}={\boldsymbol{e}}^{-1}\\\
0&{\rm others}\end{cases}\,.$

For any cycle $C=({\boldsymbol{e}}_{1},\cdots,{\boldsymbol{e}}_{k})$, which is
not necessarily primitive, we can uniquely assign a sequence of the elements
as $X_{C}\equiv X_{{\boldsymbol{e}}_{1}{\boldsymbol{e}}_{2}}\cdots
X_{{\boldsymbol{e}}_{k-1}{\boldsymbol{e}}_{k}}X_{{\boldsymbol{e}}_{k}{\boldsymbol{e}}_{1}}=q^{|C|}u^{b(C)}$.
Since $X_{C}$ is common for the equivalence class of the cycle, by restricting
the cycles to primitive cycles, we can write the Bartholdi zeta function as

$\zeta_{G}(q,u)=\prod_{C\in[{\cal P}]}(1-X_{C})^{-1}\,.$

Since we can identify the cycles with the words made of the edges, this is
precisely the situation to use (the corollary of) Amitsur’s theorem. We can
therefore write the Bartholdi zeta function (2.6) as the inverse of a
polynomial of $q$ and $u$ as announced as

$\zeta_{G}(q,u)=\det\left(\boldsymbol{1}_{2n_{E}}-q(W+uJ)\right)^{-1}\,.$
(2.8)

Of course, by setting $u=0$, the same can be concluded for the Ihara zeta
function (2.2).

The expression (2.8) is sometimes called the edge Bartholdi zeta function
because it is described through matrices $W$ and $J$ that characterize the
relation among the edges of the graph. Apart from this expression, there is
another expression focusing on the relation among the vertexes;

$\zeta_{G}(q,u)=\bigl{(}\boldsymbol{1}_{n_{V}}-(1-u)^{2}q^{2}\bigr{)}^{-(n_{E}-n_{V})}\det\bigl{(}\boldsymbol{1}_{n_{V}}-qA+(1-u)q^{2}(D-(1-u)\boldsymbol{1}_{n_{V}})\bigr{)}^{-1}\,,$
(2.9)

where $D$ is the diagonal matrix defined by $D={\rm diag}_{v\in V}(\deg(v))$
and the matrix $A$ is a square matrix of size $n_{V}$ called the vertex
adjacency matrix defined by

$A_{vv^{\prime}}=\sum_{{\boldsymbol{e}}\in
E_{D}}\delta_{\langle{v},{v^{\prime}}\rangle,{\boldsymbol{e}}}\,.\quad(v,v^{\prime}\in
V)$ (2.10)

In this paper, we will give a proof of the equivalence of (2.8) and (2.9) as a
corollary of the similar equivalence of the matrix weight Bartholdi zeta
function introduced soon below.

###  2.2 Matrix weighted Bartholdi zeta function

We here consider to put an invertible $K\times K$ matrix $X_{e}$ to each edge
$e$ of the graph. We assume that the matrix on the inverse edge $e^{-1}$ is
the inverse of the matrix on the edge $e$;

$X_{e^{-1}}=X_{e}^{-1}\,.$ (2.11)

For a cycle $C=({\boldsymbol{e}}_{i_{1}}\cdots{\boldsymbol{e}}_{i_{n}})$, we
assign a matrix,

$X_{C}\equiv X_{{\boldsymbol{e}}_{i_{1}}}\cdots X_{{\boldsymbol{e}}_{i_{n}}}.$

We then propose an extension of the Bartholdi zeta function,

$\displaystyle\zeta_{G}(q,u;X)$ $\displaystyle\equiv\prod_{C\in[{\cal
P}]}\det\bigl{(}\boldsymbol{1}_{K}-q^{|C|}u^{b(C)}X_{C}\bigr{)}^{-1}\,,$
(2.12)

which we call the matrix weighted Bartholdi zeta function in the following.
The matrix weighted Bartholdi zeta function (2.12) is an extension of the
weighted Bartholdi zeta function defined in [36, 37] where the weights on the
edges are supposed to be c-numbers. Note that it becomes the matrix weighted
Ihara zeta function introduced in [28] by taking $u=0$.

Suppose a cycle with backtracking as
${\tilde{C}}=P_{1}{\boldsymbol{e}}{\boldsymbol{e}}^{-1}P_{2}$. Then the matrix
$X_{\tilde{C}}$ reduces as
$X_{\tilde{C}}=X_{P_{1}}X_{{\boldsymbol{e}}}X_{{\boldsymbol{e}}^{-1}}X_{P_{2}}=X_{P_{1}}X_{P_{2}}$
because we have assumed $X_{{\boldsymbol{e}}^{-1}}=X_{\boldsymbol{e}}^{-1}$.
The same reduction occurs also for tails when $X_{\tilde{C}}$ is included in
the determinant as (2.12). Repeating this reduction, the matrix
$X_{{\tilde{C}}}$ finally reduces to the matrix associated with a reduced
cycle. In general, a reduced cycle is a positive power of a primitive reduced
cycle $C$. We then denote the set of representatives of primitive cycles that
are equivalent to $C^{k}$ $(k\in\mathbb{N})$ after eliminating the bumps by
$[{\cal B}(C^{k})]\subset[{\cal P}]$. We also denote the set of the
representatives of primitive cycles that reduce to a point (vertex) by
eliminating the bumps by $[{\cal B}_{0}]\subset[{\cal P}]$. From the
consideration above, $X_{\tilde{C}}$ for ${\tilde{C}}\in[{\cal B}(C^{k})]$
reduces to $X_{C}^{k}$ and thus we can rewrite (2.12) as

$\displaystyle\zeta_{G}(q,u;X)$ $\displaystyle={\cal
V}_{G}(q,u)^{K}\prod_{C\in[{\cal
P}_{R}]}\prod_{k=1}^{\infty}\prod_{{\tilde{C}}\in[{\cal
B}(C^{k})]}\det\bigl{(}\boldsymbol{1}_{K}-q^{|{\tilde{C}}|}u^{b({\tilde{C}})}X_{C}^{k}\bigr{)}^{-1}\,,$
(2.13)

where

${\cal V}_{G}(q,u)\equiv\prod_{{\tilde{C}}\in[{\cal
B}_{0}]}\frac{1}{1-q^{|{\tilde{C}}|}u^{b({\tilde{C}})}}\,.$ (2.14)

We can further evaluate it as

$\displaystyle\prod_{k=1}^{\infty}\prod_{{\tilde{C}}\in[{\cal
B}(C^{k})]}\det\left(\boldsymbol{1}_{K}-q^{|{\tilde{C}}|}u^{b({\tilde{C}})}X_{C}^{k}\right)^{-1}$
$\displaystyle=\exp\left\\{\sum_{k=1}^{\infty}\sum_{m=1}^{\infty}\frac{1}{m}\left(\sum_{{\tilde{C}}\in[{\cal
B}(C^{k})]}(q^{|{\tilde{C}}|}u^{b({\tilde{C}})})^{m}\right)\operatorname{Tr}(X_{C}^{km})\right\\}$
$\displaystyle=\exp\left\\{\sum_{n=1}^{\infty}\frac{1}{n}\sum_{k|n}\left(k\sum_{{\tilde{C}}\in[{\cal
B}(C^{k})]}(q^{|{\tilde{C}}|}u^{b({\tilde{C}})})^{n/k}\right)\operatorname{Tr}(X_{C}^{n})\right\\}$
$\displaystyle=\exp\left\\{\sum_{n=1}^{\infty}\frac{1}{n}\left(\sum_{{\tilde{C}}\in[{\cal
B}(C)]}q^{|{\tilde{C}}|}u^{b({\tilde{C}})}\right)^{n}\operatorname{Tr}(X_{C}^{n})\right\\}\,,$

where we have used the identity,

$\displaystyle\sum_{k|n}\left(k\sum_{{\tilde{C}}\in[{\cal
B}(C^{k})]}(q^{|{\tilde{C}}|}u^{b({\tilde{C}})})^{n/k}\right)=\left(\sum_{{\tilde{C}}\in[{\cal
B}(C)]}q^{|{\tilde{C}}|}u^{b({\tilde{C}})}\right)^{n}\,,$ (2.15)

which is proven in Appendix B. Therefore the matrix weighted Bartholdi zeta
function is finally written as

$\displaystyle\zeta_{G}(q,u;X)$ $\displaystyle={\cal
V}_{G}(q,u)^{K}\prod_{C\in[{\cal
P}_{R}]}\exp\left(\sum_{n=1}^{\infty}\frac{f_{C}(q,u)^{n}}{n}\operatorname{Tr}(X_{C}^{n})\right)\,,$
(2.16)

where

$f_{C}(q,u)\equiv\sum_{{\tilde{C}}\in[{\cal
B}(C)]}q^{|{\tilde{C}}|}u^{b({\tilde{C}})}\,.$ (2.17)

This expression plays an essential role in evaluating the partition function
of the KM model in the following section.

We note that the matrix weighted Bartholdi zeta function can be further
rewritten as

$\zeta_{G}(q,u;X)={\cal V}_{G}(q,u)^{K}\prod_{C\in[{\cal
P}_{R}]}\det(\boldsymbol{1}_{K}-f_{C}(q,u)X_{C})^{-1}\,,$

which can be regarded as an extension of the Ihara zeta function in the sense
that we count rather $f_{C}(q,u)X_{C}$ not $q^{|C|}$ for a primitive reduced
cycle $C$.

###  2.3 Determinant expressions of the matrix weighted Bartholdi zeta
function

As same as the original Bartholdi zeta function, the matrix weighted Bartholdi
zeta function (2.12) can be expressed as the inverse of the determinant. This
is achieved by extending the edge adjacency matrix and the bump matrix (2.7)
as

$\displaystyle(W_{X})_{{\boldsymbol{e}}{\boldsymbol{e}}^{\prime}}=\begin{cases}X_{{\boldsymbol{e}}}&{\rm
if}\ t({\boldsymbol{e}})=s({\boldsymbol{e}}^{\prime})\ {\rm and}\
{\boldsymbol{e}}^{\prime-1}\neq{\boldsymbol{e}}\\\ 0&{\rm
others}\end{cases}\,,\quad(J_{X})_{{\boldsymbol{e}}{\boldsymbol{e}}^{\prime}}=\begin{cases}X_{{\boldsymbol{e}}}&{\rm
if}\ {\boldsymbol{e}}^{\prime-1}={\boldsymbol{e}}\\\ 0&{\rm
others}\end{cases}\,.$ (2.18)

Repeating the discussion above the equation (2.8), we can show that the matrix
weighted Bartholdi zeta function can be written as

$\displaystyle\zeta_{G}(q,u;X)$
$\displaystyle=\det\bigl{(}\boldsymbol{1}_{2Kn_{E}}-q(W_{X}+uJ_{X})\bigr{)}^{-1}\,,$
(2.19)

as a direct result of Amitsur’s theorem.

We can further show that, as same as (2.9), the matrix weighted Bartholdi zeta
function can be expressed through the matrix weighted vertex adjacency matrix
of the size $Kn_{V}$,

$(A_{X})_{vv^{\prime}}=\sum_{{\boldsymbol{e}}\in
E_{D}}X_{{\boldsymbol{e}}}\,\delta_{\langle{v},{v^{\prime}}\rangle,{\boldsymbol{e}}}\,,$
(2.20)

as

$\zeta_{G}(q,u;X)=\bigl{(}1-(1-u)^{2}q^{2}\bigr{)}^{-K(n_{E}-n_{V})}\det\bigl{(}\boldsymbol{1}_{Kn_{V}}-qA_{X}+(1-u)q^{2}(D-(1-u)\boldsymbol{1}_{Kn_{V}})\bigr{)}^{-1}\,,$
(2.21)

where $D$ has been redefined as

$D_{vv^{\prime}}=\deg(v)\delta_{vv^{\prime}}\boldsymbol{1}_{K}\,.$ (2.22)

Let us prove it by using the strategy described in [38] (see also [28]):
Firstly, we define matrices $S_{X}$ and $T_{X}$ of size $n_{E}\times n_{V}$
whose elements are square matrices of size $K$;

$\displaystyle(S_{X})_{ev}\equiv\begin{cases}X_{e}^{-1}&{\rm if}\ v=s(e)\\\
0&{\rm others}\end{cases}\,,\quad(T_{X})_{ev}\equiv\begin{cases}X_{e}&{\rm
if}\ v=t(e)\\\ 0&{\rm others}\end{cases}\,.$

We also define $S\equiv(S_{X})|_{X=1}$ and $T\equiv(T_{X})|_{X=1}$. Then we
can easily show

$\displaystyle S^{T}T_{X}+T^{T}S_{X}=A_{X},\quad S^{T}S+T^{T}T=D\,.$

Using these matrices, we define

$\displaystyle L$
$\displaystyle\equiv\begin{pmatrix}\boldsymbol{1}_{Kn_{V}}&qS^{T}&qT^{T}\\\
-tqS+T_{X}&(1-t^{2}q^{2})\boldsymbol{1}_{Kn_{E}}&0\\\
-tqT+S_{X}&0&(1-t^{2}q^{2})\boldsymbol{1}_{Kn_{E}}\end{pmatrix}\,,\quad
M\equiv\begin{pmatrix}(1-t^{2}q^{2})\boldsymbol{1}_{Kn_{V}}&0&0\\\ tqS-
T_{X}&\boldsymbol{1}_{Kn_{E}}&0\\\ tqT-
S_{X}&0&\boldsymbol{1}_{Kn_{E}}\end{pmatrix}\,.$

After a straightforward calculation, we obtain

$\displaystyle LM$
$\displaystyle=\begin{pmatrix}\boldsymbol{1}_{Kn_{V}}-qA_{X}+tq^{2}(D-t\boldsymbol{1}_{Kn_{V}})&qS^{T}&qT^{T}\\\
0&(1-t^{2}q^{2})\boldsymbol{1}_{Kn_{E}}&0\\\
0&0&(1-t^{2}q^{2})\boldsymbol{1}_{Kn_{E}}\end{pmatrix}\,,$ $\displaystyle ML$
$\displaystyle=\begin{pmatrix}(1-t^{2}q^{2})\boldsymbol{1}_{Kn_{V}}&q(1-t^{2}q^{2})S^{T}\hskip
56.9055ptq(1-t^{2}q^{2})T^{T}\\\ \begin{array}[]{c}0\\\
0\end{array}&\left(\boldsymbol{1}_{2Kn_{E}}-q(W_{X}+(1-t)J_{X})\right)\left(\boldsymbol{1}_{2Kn_{E}}-tqJ_{X}\right)\end{pmatrix}\,.$

Using $\det(LM)=\det(ML)$ and ${\rm
det}\left(\boldsymbol{1}_{2Kn_{E}}-tqJ_{X}\right)=(1-t^{2}q^{2})^{Kn_{E}}$, we
find

$\qquad(1-t^{2}q^{2})^{2Kn_{E}}\det\left(\boldsymbol{1}_{Kn_{V}}-qA_{X}+tq^{2}(D-t\boldsymbol{1}_{Kn_{V}})\right)\\\
=(1-t^{2}q^{2})^{K(n_{V}+n_{E})}\det\left(\boldsymbol{1}_{2Kn_{E}}-q(W_{X}+(1-t)J_{X})\right)\,,\qquad$
(2.23)

and can conclude (2.9) by setting $t=1-u$. Note that the assumption (2.11) is
required in this computation.

##  3 Graph Kazakov-Migdal model and Bartholdi zeta function

###  3.1 The partition function of the graph Kazakov-Migdal model as Bartholdi
zeta function

The KM model [4] is a lattice gauge theory with the gauge group $SU(N)$ which
is originally defined on the $D$-dimensional square lattice with the action
(1.1). We can naturally extend it to a model on the graph. Suppose that a
Hermitian matrix $\Phi_{v}$ and a unitary matrix $U_{e}\in U(N)$ live on each
vertex $v\in V$ and each edge $e\in E$ of a given graph $G$, respectively. If
we consider gauge invariant operators of quadratic in $\Phi$ as in the
original KM model, we can consider $\operatorname{Tr}\Phi_{v}^{2}$ for each
vertex $v$ and $\operatorname{Tr}\Phi_{s(e)}U_{e}\Phi_{t(e)}U_{e}^{\dagger}$
and $\operatorname{Tr}\left(\Phi_{s(e)}^{2}+\Phi_{t(e)}^{2}\right)$ for each
edge $e$. Therefore, if we assign global coupling constants to these
operators, the most general action is given by

$\displaystyle S$
$\displaystyle=\operatorname{Tr}\left\\{\frac{m_{0}^{2}}{2}\sum_{v\in
V}\Phi_{v}^{2}+q\sum_{e\in
E}\left(\frac{r}{2}\left(\Phi_{s(e)}^{2}+\Phi_{t(e)}^{2}\right)-\Phi_{s(e)}U_{e}\Phi_{t(e)}U_{e}^{\dagger}\right)\right\\}$
$\displaystyle=\operatorname{Tr}\left\\{\frac{1}{2}\sum_{v\in
V}\left(m_{0}^{2}+qr\deg v\right)\Phi_{v}^{2}-q\sum_{e\in
E}\operatorname{Tr}\Phi_{s(e)}U_{e}\Phi_{t(e)}U_{e}^{\dagger}\right\\}\,.$
(3.1)

In the following, we fix the parameters $r$ and $m_{0}^{2}$ by using an
additional constant $u$ as $r=q(1-u)$ and $m_{0}^{2}=1-q^{2}(1-u)^{2}$,

$S_{\rm gKM}=\operatorname{Tr}\left\\{\frac{1}{2}\sum_{v\in
V}(1-q^{2}(1-u)^{2}+q^{2}(1-u)\deg v)\Phi_{v}^{2}-q\sum_{e\in
E}\Phi_{s(e)}U_{e}\Phi_{t(e)}U_{e}^{\dagger}\right\\}\,,$ (3.2)

in order to see a connection to the Bartholdi zeta function soon later.
Although this parametrization is not the most general one, the action (3.2)
reproduces the action of the original KM model (1.1) by setting the graph $G$
to the $D$-dimensional square lattice and tuning $q$ and $u$ appropriately. We
call the model with the action (3.2) the graph Kazakov-Migdal (gKM) model.
Note that the model considered in [28] is obtained by setting $u=0$.

We next consider the partition function of the gKM model,

$Z_{\rm gKM}=\int\prod_{v\in V}d\Phi_{v}\prod_{e\in E}dU_{e}\,e^{-\beta S_{\rm
gKM}}\,.$ (3.3)

As discussed in [4] (see also [28]), there are two different representations
of the partition function $Z_{\rm gKM}$ depending on whether the scalar fields
$\Phi_{v}$ is integrated first or the gauge fields $U_{e}$ is integrated
first. Let us first consider the case of integrating the scalar fields
$\Phi_{v}$ first. The action (3.2) is bilinear in the scalar fields $\Phi_{v}$
and can be written as

$S_{\rm
gKM}=\frac{1}{2}\Phi_{v,a}\left(\boldsymbol{1}_{n_{V}N^{2}}-qA_{U}+q^{2}(1-u)(D-(1-u)\boldsymbol{1}_{n_{V}N^{2}})\right)_{va,v^{\prime}b}\Phi_{v^{\prime}b}\,,$
(3.4)

where $a,b=1,\cdots,N^{2}$ and $A_{U}$ is the adjacency matrix given by (2.20)
with $X_{e}=U_{e}\otimes U_{e}^{\dagger}$ and $D$ is the degree matrix defined
in (2.22). The matrix that appeared in this expression is the same as the one
that appeared in the vertex representation of the matrix weighted Bartholdi
zeta function (2.21). Therefore the integration over the scalar fields
$\Phi_{v}$ in the partition function yields

$\displaystyle Z_{\rm gKM}$
$\displaystyle=\left(\frac{2\pi}{\beta}\right)^{\frac{1}{2}n_{V}N^{2}}\int\prod_{e\in
E}dU_{e}\,{\det}\left({\boldsymbol{1}_{n_{V}N^{2}}-qA_{U}+q^{2}(1-u)(D-(1-u)\boldsymbol{1}_{n_{V}})}\right)^{-\frac{1}{2}}$
$\displaystyle=\left(\frac{2\pi}{\beta}\right)^{\frac{1}{2}n_{V}N^{2}}(1-(1-u)^{2}q^{2})^{\frac{1}{2}{(n_{E}-n_{V})N^{2}}}\int\prod_{e\in
E}dU_{e}\,\zeta_{G}(q,u;U)^{\frac{1}{2}}$
$\displaystyle=\left(\frac{2\pi}{\beta}\right)^{\frac{1}{2}n_{V}N^{2}}(1-(1-u)^{2}q^{2})^{\frac{1}{2}{(n_{E}-n_{V})N^{2}}}{\cal
V}_{G}(q,u)^{\frac{N^{2}}{2}}$ $\displaystyle\hskip
113.81102pt\times\int\prod_{e\in
E}dU_{e}\,\prod_{C\in\Pi^{+}}e^{\sum_{m=1}^{\infty}\frac{1}{m}f_{C}(q,u)^{m}|\operatorname{Tr}P_{C}(U)^{m}|^{2}}$
(3.5)

where $\zeta_{G}(q,u;U)$ is the matrix weighted Bartholdi zeta function (2.12)
with $X_{e}=U_{e}\otimes U_{e}^{\dagger}$ and we have used (2.16) in the last
line. We note that, using the left-right invariance of the Haar measure
$\prod_{e\in E}dU_{e}$ in (3.3) and the gauge invariance of the action (3.2),
we can fix $U_{e}=1$ on the edges of a spanning tree of the graph $G$, that
is, a subgraph of $G$ that contains all the vertices of $G$ and has no cycles.

###  3.2 The dual description of the partition function and a relation to
covering graphs

We next consider integrating over the gauge fields $U_{e}$ first. To this end,
we need the so-called Harish-Chandra-Itzykson-Zuber (HCIZ) integral formula
[39];

$\int
dU\,e^{t\operatorname{Tr}AUBU^{\dagger}}=\frac{G(N+1)}{t^{\frac{N^{2}-N}{2}}}\frac{{\det}_{i,j}\left({e^{ta_{i}b_{j}}}\right)}{\Delta(a)\Delta(b)}\,,$
(3.6)

where $U$ is a unitary matrix of size $N$, $dU$ is the Haar measure of $U(N)$
normalized as $\int dU=1$, $G(N+1)=\prod_{i=1}^{N-1}i!$ is the Barnes double
gamma function, $A$ and $B$ are Hermitian matrices whose eigenvalues are
$(a_{1},\cdots,a_{N})$ and $(b_{1},\cdots,b_{N})$, respectively, and
$\Delta(a)$ and $\Delta(b)$ are the Vandermonde determinant with respect to
$A$ and $B$, respectively;

$\Delta(a)=\prod_{1\leq i<j\leq N}(a_{j}-a_{i})\,,\quad\Delta(b)=\prod_{1\leq
i<j\leq N}(b_{j}-b_{i})\,.$

We apply the formula (3.6) to the partition function of the gKM model (3.3).
Since the integrand includes only the eigenvalues of the Hermitian matrices
$\Phi_{v}$, we also use a mapping from the matrix integral to the integral
over the eigenvalues (see e.g.[40]),

$\int d\Phi
f(\Phi)=\frac{(2\pi)^{\frac{N(N-1)}{2}}}{G(N+2)}\int\prod_{i=1}^{N}d\phi_{i}\,\Delta(\phi)^{2}f(\phi)\,,$

for a Hermitian matrix $\Phi$ with eigenvalues $\phi_{i}$ ($i=1,\cdots,N$). We
then obtain

$\displaystyle Z_{\rm gKM}$ $\displaystyle={\cal N}\int\left({\prod_{v\in
V}\prod_{i=1}^{N}d\phi_{v,i}}\right)\left({\prod_{v^{\prime}\in
V}\Delta(\phi_{v^{\prime}})^{2-\deg
v^{\prime}}}\right)\sum_{\sigma_{1},\cdots,\sigma_{n_{E}}\in S_{N}}{\rm
sgn}\,(\sigma_{1}\cdots\sigma_{n_{E}})e^{-\frac{1}{2}\phi_{v,i}(D_{\sigma})_{vi;v^{\prime}j}\phi_{v^{\prime},j}}\,,$
(3.7)

where

${\cal
N}\equiv\frac{{G(N+1)}^{n_{E}-n_{V}}}{(N!)^{n_{V}}}\frac{(2\pi)^{\frac{N(N-1)}{2}n_{V}}}{\beta^{\frac{N^{2}}{2}n_{V}}q^{\frac{N(N-1)}{2}n_{E}}}\,,$

and

$(D_{\sigma})_{vi;v^{\prime}j}\equiv\Bigl{(}1+(1-u)q^{2}(\deg
v-(1-u))\Bigr{)}\delta_{vv^{\prime}}\delta_{ij}-qA_{vv^{\prime}}\Bigl{(}\delta_{i,\sigma_{\langle
vv^{\prime}\rangle}^{-1}(j)}+\delta_{j,\sigma_{\langle
vv^{\prime}\rangle}(i)}\Bigr{)}\,,$ (3.8)

with the vertex adjacency matrix $A$ given in (2.10).

It is remarkable that the matrix (3.8) is the matrix appeared in the vertex
representation of the Bartholdi zeta function (2.9) corresponding to a
covering graph of $G$: Imagine that $N$ copies of the graphs $G$ are stacked
with their vertices aligned. We label the vertex $v$ of the $i$’s layer as
$(v,i)$. Let $\langle v,v^{\prime}\rangle$ be an edge of the original graph
$G$, and we consider to reconnect the $N$ vertices above $v$ and $v^{\prime}$
by edges in a one-to-one relationship. If we connect the vertices in the same
layer $(v,i)$ and $(v^{\prime},i)$ ($i=1,\cdots,N$) for all $\langle
v,v^{\prime}\rangle\in E$, it becomes simply $N$ copies of the original graph
$G$. Instead, we can connect vertices in different layers, $(v,i)$ and
$(v^{\prime},j)$ $(i\neq j)$. In this case, by making such recombination on
every edge, we can create a graph that is different from $G$, which is called
a covering graph of $G$. Note that the obtained covering graph is disconnected
in general. Since which layers are connected at edge $e$ is specified by the
permutation of $N$, the vertex adjacency matrix of the covering graph is
expressed as a direct product of the vertex adjacency matrix $A$ of the
original graph and elements of $S_{N}$. It is exactly the one that appeared in
the second term of (3.8). From the construction, the degree of the vertex
$(v,i)$ of the covering graph is the same with the degree $v$ of the original
graph $G$. Therefore we see that (3.8) is exactly the matrix that appeared in
the vertex representation of the Bartholdi zeta function (2.9) for the
covering graph.

We note that there is a residual gauge symmetry in the expression (3.7) to
permute the eigenvalues of $\Phi_{v}$ as $\phi_{v,i}\to\phi_{v,\tau_{v}(i)}$
by ${}^{\forall}\tau_{v}\in S_{N}$, which transforms the permutation
$\sigma_{e}$ on the edge $e$ as
$\sigma_{e}\to\tau_{s(e)}\sigma_{e}\tau_{t(e)}$. Therefore, we can fix this
gauge by setting the permutations on the edges of a spanning tree of the graph
as $\sigma_{e}=1$. This symmetry corresponds to the freedom to swap
permutations that create the same covering graph. We will see it in a concrete
example in the next section.

###  3.3 Graph Kazakov-Migdal model on a $d$-regular graph

We next restrict the graph to a $d$-regular graph, that is, a graph where each
vertex has the same degree $d$. In this case, the action of the gKM model
(3.1) can be written in the same manner with the original KM model (1.1) as

$S_{\rm gKM}=q\operatorname{Tr}\left\\{\frac{m^{2}}{2}\sum_{v\in
V}\Phi_{v}^{2}-\sum_{e\in
E}\Phi_{s(e)}U_{e}\Phi_{t(e)}U_{e}^{\dagger}\right\\}\,,$ (3.9)

where the mass parameter $m^{2}$ is related to the parameters $q$ and $u$ as

$m^{2}=q^{-1}+q\bigl{(}(1-u)d-(1-u)^{2}\bigr{)}\,.$ (3.10)

Note that, although $q$ can be absorbed into $\beta$ in the definition of the
partition function (3.3), we have kept it to make the relation to the original
parametrization clearer.

In this parametrization, we can write the partition function as

$\displaystyle Z_{\rm gKM}$ $\displaystyle=\left(\frac{2\pi}{\beta
q\,m^{2}}\right)^{\frac{1}{2}n_{V}N^{2}}\int\prod_{e\in
E}dU_{e}\,{\det}\left({\boldsymbol{1}_{n_{V}N^{2}}-m^{-2}A_{U}}\right)^{-\frac{1}{2}}$
$\displaystyle=\left(\frac{2\pi}{\beta
q\,m^{2}}\right)^{\frac{1}{2}n_{V}N^{2}}\int\prod_{e\in
E}dU_{e}\,\exp\left(\frac{1}{2}\sum_{k=1}^{\infty}\frac{m^{-2k}}{k}\operatorname{Tr}A_{U}^{k}\right)$
$\displaystyle=\left(\frac{2\pi}{\beta
q\,m^{2}}\right)^{\frac{1}{2}n_{V}N^{2}}\int\prod_{e\in
E}dU_{e}\,\exp\left(\frac{1}{2}\sum_{k=1}^{\infty}\frac{m^{-2k}}{k}\sum_{|C|=k}\bigl{|}\operatorname{Tr}P_{C}(U)\bigr{|}^{2}\right)\,,$
(3.11)

where we have used the fact that $\operatorname{Tr}A_{U}^{k}$ counts all
Wilson loops with length $k$ and the summation of the cycles in the
parenthesis of the last line runs over all cycles with length $k$. Since any
cycle can be written as a positive power of a primitive cycle and the set of
the representatives of primitive cycles $[{\cal P}]$ can be decomposed into
$[{\cal B}_{0}]$ and $[{\cal B}({C}^{j})]$ (${C}\in[{\cal P}_{R}]$,
$j\in\mathbb{N}$), we can estimate the expression in the parenthesis as (see
also Fig. 1.)

$\displaystyle\frac{1}{2}$
$\displaystyle\sum_{k=1}^{\infty}\frac{m^{-2k}}{k}\sum_{|C|=k}\bigl{|}\operatorname{Tr}P_{C}(U)\bigr{|}^{2}$
$\displaystyle=\frac{1}{2}\sum_{C\in[{\cal
P}]}|C|\sum_{l=1}^{\infty}\frac{m^{-2l|C|}}{l|C|}\bigl{|}\operatorname{Tr}P_{C}(U)^{l}\bigr{|}^{2}$
$\displaystyle=\frac{N^{2}}{2}\sum_{{\tilde{C}}\in[{\cal
B}_{0}]}\sum_{l=1}^{\infty}\frac{m^{-2l|{\tilde{C}}|}}{l}+\sum_{{C}\in[{\cal
P}_{R}]}\sum_{j=1}^{\infty}\sum_{{\tilde{C}}\in[{\cal
B}({C}^{j})]}\sum_{l=1}^{\infty}\frac{m^{-2l|{\tilde{C}}|}}{l}\bigl{|}\operatorname{Tr}P_{{C}}(U)^{jl}\bigr{|}^{2}$
$\displaystyle=-\frac{N^{2}}{2}\sum_{{\tilde{C}}\in[{\cal
B}_{0}]}\log\left(1-m^{-2|{\tilde{C}}|}\right)+\sum_{{C}\in[{\cal
P}_{R}]}\sum_{n=1}^{\infty}\frac{1}{n}\left(\sum_{{\tilde{C}}\in[{\cal
B}({C})]}m^{-2|{\tilde{C}}|}\right)^{n}\bigl{|}\operatorname{Tr}P_{{C}}(U)^{n}\bigr{|}^{2}\,,$

where we have used the fact that the equivalence class of the cycle $C^{l}$
$(C\in[{\cal P}])$ has $|C|$ different elements in the first line and we have
repeated the same calculation above (2.15) to obtain the last line. We can
then rewrite (3.11) as

$\displaystyle Z_{\rm gKM}$ $\displaystyle=\left(\frac{2\pi}{\beta
q\,m^{2}}\right)^{\frac{1}{2}n_{V}N^{2}}{\cal
V}_{G}(m^{-2},1)^{\frac{N^{2}}{2}}\int\prod_{e\in
E}dU_{e}\,\exp\left(\sum_{{C}\in[\Pi^{+}]}\sum_{n=1}^{\infty}\frac{1}{n}f_{{C}}(m^{-2},1)^{n}\bigl{|}\operatorname{Tr}P_{{C}}(U)^{n}\bigr{|}^{2}\right)\,,$
(3.12)

where ${\cal V}_{G}(q,u)$ and $f_{C}(q,u)$ are defined in (2.14) and (2.17),
respectively.

(a) The zero area Wilson loop on a cycle in ${\cal B}_{0}$.

(b) The non-zero area Wilson loop on a cycle in ${\cal B}(C)$.

Figure 1:  Examples of the Wilson loops on the cycles with branches and bumps.
The Wilson loop on the cycle (a) reduces to a point (vertex) and $P_{C}(U)=1$
since $UU^{\dagger}=1$ on round trips over the edges. For the same reason, the
Wilson loop on the cycle (b) gives the same contribution to the Wilson loop on
the cycle $C=(e_{1},e_{2},e_{3},e_{4},e_{5})$ without branches and bumps
(dashed cycle).

Comparing this expression to (3.5), we obtain non-trivial identities satisfied
by $d$-regular graphs:

$\displaystyle\begin{split}(qm^{2})^{-n_{V}}{\cal
V}_{G}(m^{-2},1)&=(1-(1-u)^{2}q^{2})^{n_{E}-n_{V}}{\cal V}_{G}(q,u)\,,\\\
f_{C}(m^{-2},1)&=f_{C}(q,u)\,.\end{split}$ (3.13)

In particular the relation ${\cal V}_{G}(q,0)=1$ and $f_{C}(q,0)=q^{|C|}$
yields the identities,

$\displaystyle\begin{split}\prod_{{\tilde{C}}\in[{\cal
B}_{0}]}\left(1-(q^{-1}+(d-1)q)^{|{\tilde{C}}|}\right)^{-1}&=\bigl{(}(1+(d-1)q^{2})(1-q^{2})^{d-2}\bigr{)}^{n_{V}}\,,\\\
\sum_{{\tilde{C}}\in[{\cal
B}(C)]}(q^{-1}+(d-1)q)^{-|{\tilde{C}}|}&=q^{|C|}\,.\end{split}$ (3.14)

##  4 Exact partition function of the graph Kazakov-Migdal model on cycle
graphs at finite $N$

In [28], the integral (3.7) with $u=0$ for the cycle graph $C_{n}$, that is, a
connected graph that has $n$ vertices and $n$ edges and $\deg v=2$ for
${}^{\forall}v\in V$, was evaluated exactly at finite $N$. In this section, we
show that the same can be performed for the gKM model in general parameters.

Figure 2: A cycle graph $C_{n}$ with $n$ vertices and edges

###  4.1 Evaluation by unitary matrix integral

Let us first evaluate the Bartholdi zeta function for $C_{n}$ using the
expression (2.9);

$\displaystyle\zeta_{C_{n}}(q,u)=(\det M_{n})^{-1}\,,$ (4.1)

where the matrix $M_{n}$ is defined by

$\displaystyle M_{1}$ $\displaystyle=1-2q+(1-u^{2})q^{2},\quad
M_{2}=\begin{pmatrix}1+(1-u^{2})q^{2}&-2q\\\
-2q&1+(1-u^{2})q^{2}\end{pmatrix},$ $\displaystyle M_{n}$
$\displaystyle=\begin{pmatrix}1+(1-u^{2})q^{2}&-q&\cdots&0&-q\\\
-q&1+(1-u^{2})q^{2}&\cdots&0&0\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&\cdots&1+(1-u^{2})q^{2}&-q\\\
-q&0&\cdots&-q&1+(1-u^{2})q^{2}\end{pmatrix}\ (n\geq 3)\,.$ (4.2)

It is straightforward to evaluate the determinant and the result is

$\zeta_{C_{n}}(q,u)=\Bigl{(}\xi_{+}(q,u)^{\frac{n}{2}}-\xi_{-}(q,u)^{\frac{n}{2}}\Bigr{)}^{-2}\,,$
(4.3)

where $\xi_{\pm}$ are the solutions of the quadratic equation
$x^{2}-(1+(1-u^{2})q^{2})x+q^{2}=0$;

$\xi_{\pm}(q,u)\equiv\frac{1}{2}\Bigl{(}1+(1-u^{2})q^{2}\pm\sqrt{1-2(1+u^{2})q^{2}+(1-u^{2})^{2}q^{4}}\Bigr{)}\,.$
(4.4)

We next evaluate the partition function $Z_{\rm gKM}$ for $C_{n}$ by using the
expression (3.5). Since excluding the edge $e=1$ from $C_{n}$ yields the
spanning graph of $C_{n}$, we can fix the gauge by setting
$U_{2}=\cdots=U_{n}=1$ and the gauge field remains only on the first edge
$e=1$. We then denote $U_{1}=U$ in the following. The matrix weighted
Bartholdi zeta function $\zeta_{C_{n}}(q,u;U)$ in this gauge can be computed
by using the same technique deriving (4.3) as333In this computation, we have
also used
$\frac{1}{2q^{2n}}\left(2q^{n}+\zeta_{C_{n}}(q,u)^{-1}-\sqrt{\zeta_{C_{n}}(q,u)^{-2}+4q^{n}\zeta_{C_{n}}(q,u)^{-1}}\right)=\xi_{+}(q,u)^{-n}\,.$

$\displaystyle\zeta_{C_{n}}(q,u;U)$
$\displaystyle=\det\Bigl{(}\zeta_{C_{n}}(q,u)^{-1}\boldsymbol{1}_{N^{2}}+(2-U\otimes
U^{\dagger}-U^{\dagger}\otimes U)q^{n}\Bigr{)}^{-1}$
$\displaystyle=\xi_{+}(q,u)^{-N^{2}n}\Bigl{|}\det\Bigl{(}\boldsymbol{1}_{N^{2}}-q^{n}\xi_{+}(q,u)^{-n}\,U\otimes
U^{\dagger}\Bigr{)}\Bigr{|}^{-2}$
$\displaystyle=\xi_{+}(q,u)^{-N^{2}n}e^{2\sum_{k=1}^{\infty}\frac{1}{k}(q^{n}\xi_{+}(q,u)^{-n})^{k}|\operatorname{Tr}U^{k}|^{2}}\,,$
(4.5)

where we have used $\zeta_{C_{n}}(q,u)$ is expressed as (4.3). Substituting
(4.5) to (3.5) we obtain

$\displaystyle Z_{\rm gKM}$
$\displaystyle=\left(\frac{2\pi}{\beta}\right)^{\frac{N^{2}}{2}n}\xi_{+}(q,u)^{-\frac{N^{2}}{2}n}\prod_{i=1}^{N}\Bigl{(}{1-q^{ni}\xi_{+}(q,u)^{-ni}}\Bigr{)}^{-1}\,,$
(4.6)

where we have used the formula obtained in [28],

$\displaystyle\int
dU\,e^{\sum_{k=1}^{\infty}\frac{q^{k}}{k}|\operatorname{Tr}U^{k}|^{2}}$
$\displaystyle=\prod_{i=1}^{N}\frac{1}{1-q^{i}}\,,\quad\text{for }|q|<1.$
(4.7)

Note that it is easy to see $\xi_{+}(q,0)=1$, which reproduces the result
obtained in [28].

###  4.2 Derivation using the covering graphs

We can derive (4.6) also from the dual expression (3.7). In this case, the
Bartholdi zeta functions of the covering graphs play an important role instead
of the unitary matrix integral.

For $G=C_{n}$, we can fix the gauge by setting
$\sigma_{e=2}=\cdots=\sigma_{e=n}=1$ and we write $\sigma_{1}=\sigma$. Then
the partition function (3.7) becomes a summation of Gaussian integrals,

$\displaystyle Z_{\rm gKM}$
$\displaystyle=\frac{1}{N!}\frac{(2\pi)^{\frac{N(N-1)}{2}n}}{\beta^{\frac{N^{2}}{2}n}q^{\frac{N(N-1)}{2}n}}\int\left({\prod_{v=1}^{n}\prod_{i=1}^{N}d\phi_{v,i}}\right)\sum_{\sigma_{\in}S_{N}}{\rm
sgn}\,(\sigma)e^{-\frac{1}{2}\phi_{v,i}(D_{\sigma})_{vi;v^{\prime}j}\phi_{v^{\prime},j}}\,.$
(4.8)

As mentioned below (3.8), $D_{\sigma}$ is the matrix that associated with the
Bartholdi zeta function (2.9) of a covering graph of $C_{n}$. In this case,
the recombination of the edges occurs only at the edge $e=1$ by the
permutation $\sigma$. The permutation is a product of cyclic permutations in
general and the elements of $S_{N}$ can be classified by a conjugacy class
labeled by the number and the lengths of the cyclic permutations, that is, a
partition of $N$. Let us write a partition of $N$ as
$\lambda=(l_{1}^{m_{1}},\cdots,l_{r}^{m_{r}})$ where $m_{i}$ is the number of
the cyclic permutations of the length $l_{i}$. From the construction, a cyclic
permutation of the length $l$ connects $l$ layers of $C_{n}$ in turn which
yields a cyclic graph $C_{ln}$. Therefore the covering graph appeared by a
permutation $\sigma$ in the conjugacy class labeled by $\lambda$ is
$(C_{nl_{1}})^{m_{1}}\oplus\cdots\oplus(C_{nl_{r}})^{m_{r}}$ and we see

$(\det D_{\sigma})^{-1}=\prod_{i=1}^{r}\zeta_{C_{nl_{i}}}(q,u)^{m_{i}}\,.$
(4.9)

Recalling that there are $N!/z_{\lambda}$ elements in the conjugacy class
labeled by $\lambda$ with

$z_{\lambda}\equiv\prod_{i=1}^{r}m_{i}!\,l_{i}^{m_{i}}\,,$ (4.10)

and the signature of the permutation is written as ${\rm sgn}\
\sigma=\prod_{i=1}^{r}(-1)^{(l_{i}-1)m_{i}}=(-1)^{N}\prod_{i=1}^{r}(-1)^{m_{i}}$,
we can perform the Gaussian integral in (4.8) as

$\displaystyle Z_{\rm gKM}$
$\displaystyle=\frac{1}{N!}\left(\frac{2\pi}{\beta}\right)^{\frac{N^{2}}{2}n}\frac{(-1)^{N}}{q^{\frac{N(N-1)}{2}n}}\sum_{\lambda\vdash
N}\prod_{i=1}^{r}\frac{N!}{z_{\lambda}}\left(-\zeta_{C_{nl_{i}}}(q,u)^{1/2}\right)^{m_{i}}$
$\displaystyle=\left(\frac{2\pi}{\beta}\right)^{\frac{N^{2}}{2}n}\frac{(-1)^{N}}{q^{\frac{N(N-1)}{2}n}}\sum_{\lambda\vdash
N}\prod_{i=1}^{r}\frac{1}{z_{\lambda}}\left(\frac{1}{\xi_{-}(q,r)^{\frac{nl_{i}}{2}}-\xi_{+}(q,r)^{\frac{nl_{i}}{2}}}\right)^{m_{i}}$
$\displaystyle=\left(\frac{2\pi}{\beta}\right)^{\frac{N^{2}}{2}n}\frac{(-1)^{N}}{q^{\frac{N(N-1)}{2}n}\xi_{-}(q,u)^{\frac{Nn}{2}}}\sum_{\lambda\vdash
N}\prod_{i=1}^{r}\frac{1}{z_{\lambda}}\left(\frac{1}{1-(\xi_{+}(q,u)^{\frac{n}{2}}/\xi_{-}(q,u)^{\frac{n}{2}})^{l_{i}}}\right)^{m_{i}}\,,$
(4.11)

where we have used (4.3) and $\xi_{-}\leq\xi_{+}$ from the first line to the
second line. To evaluate the summation in (4.11), we can use the identity,
(see e.g. [28] for a proof)

$\sum_{\lambda\vdash
N}\prod_{i=1}^{r}\frac{1}{z_{\lambda}}\left(\frac{1}{1-x^{l_{i}}}\right)^{m_{i}}=\prod_{i=1}^{N}\frac{1}{1-x^{i}}\,.$
(4.12)

Combining $\xi_{-}(q,u)\xi_{+}(q,u)=q^{2}$, we finally obtain

$\displaystyle Z_{\rm gKM}$
$\displaystyle=\left(\frac{2\pi}{\beta}\right)^{\frac{N^{2}}{2}n}\xi_{+}(q,u)^{-\frac{N^{2}}{2}n}\prod_{i=1}^{N}\Bigl{(}{1-q^{ni}\xi_{+}(q,u)^{-ni}}\Bigr{)}^{-1}\,,$

for $C_{n}$, which is identical with (4.6).

###  4.3 Counting the cycles and the Catalan numbers

The expression (4.6) is the realization of (2.16) for $G=C_{n}$, namely ${\cal
V}_{C_{n}}(q,u)=\xi_{+}(q,u)^{-n}$ and $f_{C}(q,u)=q^{n}\xi_{+}(q,u)^{-n}$. We
can explicitly see that the function $\xi_{+}(q,u)^{-n}$ and
$q^{n}\xi_{+}(q,u)^{-n}$ count such cycles that reduce to a vertex (we refer
to them as “collapsing cycles” in the following) and the primitive reduced
cycle $C$ of $C_{n}$, respectively.

Let us first consider the case of $n=1$. The point is that the function

$\xi_{+}(q,u)^{-1}=\frac{\xi_{-}(q,u)}{q^{2}}=\frac{1+(1-u^{2})q^{2}-\sqrt{1-2(1+u^{2})q^{2}+(1-u^{2})^{2}q^{4}}}{2q^{2}}\,,$
(4.13)

is the generating function of the generalized Catalan numbers [41]. The usual
Catalan number ${\rm Cat}(m)$ can be interpreted as the number of the Dyck
words of length $2m$, that is, such sequences of two letters $X$ and $Y$ of
the same number $(m)$ such that no initial segment of the sequence has more
$X$ than $Y$, which appears as the coefficient of $q^{2m}$ when expanding the
expression (4.13) by $q$ with $u=1$. The generalized Catalan number also
counts the number of the borders of $X$’s and $Y$’s in the Dyck words, which
appears in the coefficient of $u$ in the expansion of (4.13). This is exactly
the way to construct collapsing primitive cycles from the two primitive
reduced cycles $C$ and $C^{-1}$ of $G=C_{1}$. Since the powers of $q$ and $u$
can be naturally interpreted as the length of the cycle and the number of the
bumps, respectively, we can conclude that $\xi_{+}(q,u)^{-1}$ counts the
collapsing cycles. The same is true with respect to the element of $[{\cal
B}_{C}]$ and $q\xi_{+}(q,u)^{-1}$ counts the cycles in $[{\cal B}_{C}]$, since
any cycle ${\tilde{C}}\in[{\cal B}_{C}]$ is constructed by putting a
collapsing primitive cycle to $C$.

We can generalize this argument to an arbitrary $n$: The collapsing cycles of
$G=C_{n}$ can be again counted by Dyck words. To see it, we write the $n$
edges of $G=C_{n}$ as $e_{i}$ ($i=1,\cdots,n$). A path of $C_{n}$ can be
expressed as a sequence of $e_{i}$, but there is a strict rule for the order
of the sequence: After $e_{i}$, only $e_{i+1}$ or $e_{i}^{-1}$ can appear, and
after $e_{i}^{-1}$, only $e_{i}$ or $e_{i-1}^{-1}$ can appear. Therefore, if
we fix the starting point of the path, we can omit the subscript $i$ of the
edges to express the path. As a result, the path is uniquely specified as the
starting point and a sequence of $e$ and $e^{-1}$. In particular, if the path
is a collapsing cycle, it is expressed as a Dyck word by properly rotating the
cycle if necessary. Since $C_{n}$ has $n$ vertices, we see ${\cal
V}_{C^{n}}(q,u)=\xi_{+}(q,u)^{-n}$.

We next consider a path ${\tilde{C}}\in[{\cal B}_{C}]$ with
$C=(e_{1},\cdots,e_{n})$. The key observation is that ${\tilde{C}}$ always
contains the pairs $e_{i}e_{i+1}$ ($i=1,\cdots,n$). Therefore, by rotating the
order of $e_{i}$’s properly, we can express ${\tilde{C}}$ as

${\tilde{C}}=(w_{1},e_{1},w_{2},e_{2},\cdots,w_{n},e_{n})\,,$

where $w_{i}$ is a collapsing cycle starting from $e_{i}$ and not including
the pair $e_{i}e_{i+1}$. As mentioned above, we can omit the indices of the
edges to express the path. Using this rule, ${\tilde{C}}$ corresponds to a
word made of $e$ and $e^{-1}$,

$d_{1}(e,e^{-1})e\,d_{2}(e,e^{-1})e\,\cdots d_{n}(e,e^{-1})e\,,$

where $d_{i}(e,e^{-1})$ is a Dyck word made of $e$ and $e^{-1}$, which comes
from $w_{i}$. Therefore, there is one-to-one correspondence between the
element of $[{\cal B}_{C}]$ and $n$ set of Dyck words, which yields
$f_{C}(q,u)=q^{n}\xi_{+}(q,u)^{-n}$.

###  4.4 The partition function in the parametrization of the original KM
model

Although the parametrization of the gKM model adopted in (3.2) is convenient
to see the relation between the gKM model and the Bartholdi zeta function, it
is worth looking at how the partition function appears in the parametrization
of the original KM model, since the cycle graph $C_{n}$ we are considering is
nothing but the one-dimensional square lattice where the exact form of the
partition function is known [10]. We then consider the action (3.9) with $d=2$
and the mass parameter (3.10) becomes

$m^{2}=q^{-1}+q(1-u^{2})\,.$ (4.14)

This reparametrization makes the expressions simpler. Since the generating
function of the generalized Catalan numbers (4.13) can be rewritten as

$\xi_{+}(q,u)^{-1}=\frac{R(m^{2})}{q}\,,$ (4.15)

where

$R(m^{2})\equiv\frac{m^{2}-\sqrt{m^{4}-4}}{2}\,,$ (4.16)

the partition function (4.6) can be written as

$\displaystyle Z_{\rm gKM}$ $\displaystyle=\left(\frac{2\pi}{\beta
q}\right)^{\frac{N^{2}}{2}n}R(m^{2})^{\frac{N^{2}}{2}n}\prod_{i=1}^{N}\Bigl{(}{1-R(m^{2})^{ni}}\Bigr{)}^{-1}\,,$
(4.17)

for the cycle graph $C_{n}$, which reproduces the result in [10]. We can
explicitly see that the identities (3.13) (and (3.14)) hold.

We point out that the function $R(m^{2})$ is the Stieltjes transformation of
the semi-circle distribution (or the expectation value of the resolvent);

$R(m^{2})=\frac{1}{2\pi}\int_{-2}^{2}dx\frac{\sqrt{4-x^{2}}}{m^{2}-x}\,.$
(4.18)

It is plausible to think that this is related to the result [6] that the semi-
circle distribution is the exact solution of the eigenvalue distribution of
the scalar fields of the original KM model in large N (see Sect. 5). It is
interesting, however, that the function representing the semi-circle
distribution already appears in the partition function at finite $N$, which is
a different situation from the Gaussian matrix model. This reflects the fact
that the one-dimensional KM model cannot be completely regarded as a Gaussian
matrix model. In fact, as already mentioned, we cannot fix all unitary
matrices of a one-dimensional KM model but there always remains a unitary
matrix on one edge. This is why the partition function (4.6) or (4.17) does
not turn out to be that of the Gaussian matrix model.

Another insight obtained by this reparametrization is the relation between the
Catalan numbers and the generalized Catalan numbers: Recalling that the
Catalan numbers are obtained as the moments of the semi-circle distribution,

$\frac{1}{2\pi}\int_{-2}^{2}dx\,x^{k}\sqrt{4-x^{2}}=\begin{cases}{\rm
Cat}\left(\frac{k}{2}\right)&\text{$k$: even}\\\ 0&\text{$k$: odd}\\\
\end{cases}\,,$ (4.19)

we can rewrite the generating function (4.13) as

$\xi_{+}(q,u)^{-1}=\frac{1}{q}\sum_{k=0}^{\infty}\frac{{\rm
Cat}(k)}{(m^{2})^{2k+1}}=\frac{1}{q}\sum_{k=0}^{\infty}\frac{{\rm
Cat}(k)}{(q^{-1}+q(1-u^{2}))^{2k+1}}\,,$ (4.20)

where we have used (4.14). This gives a direct relation between the Catalan
numbers and the generalized Catalan numbers.

##  5 Graph Kazakov-Migdal model at large $N$

###  5.1 Exact evaluation of the partition function

We next evaluate the partition function (3.3) for an arbitrary graph $G$ in
the limit of $N\to\infty$ by carrying out the unitary matrix integral of
(3.5). To this end, we denote the Wilson loop along a primitive reduced cycle
$C$ of the graph as $P_{C}(U)$, and introduce the notations,

$|\operatorname{Tr}f(U)|^{2}\equiv\operatorname{Tr}f(U)\,\operatorname{Tr}f(U)^{\dagger}\,,$

and

$\Bigl{\langle}g(U)\Bigr{\rangle}\equiv\int\prod_{e\in E}dU_{e}\,g(U)\,,$

where $f(U)=f(U_{1},\cdots,U_{n_{E}})$ and $g(U)=g(U_{1},\cdots,U_{n_{E}})$
are functions of $U_{e}$ ($e=1,\cdots,n_{E}$).

Although it seems impossible at first glance to perform the complex integral
involving multiple unitary matrices, a significant simplification occurs at
large $N$ [42, 43, 44]. The essential facts are that the integral of
$|\operatorname{Tr}P_{C}(U)^{l}|^{2}$ at large $N$ as

$\Bigl{\langle}|\operatorname{Tr}P_{C}(U)^{l}|^{2}\Bigr{\rangle}\underset{N\to\infty}{\longrightarrow}l\,,$
(5.1)

and that the integral of the product of the two Wilson loops along two (not
necessarily primitive) cycles $C_{1}$ and $C_{2}$ are decomposed into the
product of the integrals of the individual Wilson loops at large $N$ as

$\displaystyle\Bigl{\langle}\bigl{|}\operatorname{Tr}P_{C_{1}}(U)\operatorname{Tr}P_{C_{2}}(U)\bigr{|}^{2}\Bigr{\rangle}\underset{N\to\infty}{\longrightarrow}\Bigl{\langle}\bigl{|}\operatorname{Tr}P_{C_{1}}(U)\bigr{|}^{2}\Bigr{\rangle}\Bigl{\langle}\bigl{|}\operatorname{Tr}P_{C_{2}}(U)\bigr{|}^{2}\Bigr{\rangle}\,.$
(5.2)

In general, we can define a general Wilson loop along a primitive cycle $C$
associated with a partition $\lambda=(l_{1}^{m_{1}}l_{2}^{m_{2}}\cdots)$ by

$\Upsilon_{\lambda}(P_{C}(U))\equiv\prod_{i}(\operatorname{Tr}P_{C}(U)^{l_{i}})^{m_{i}}\,.$

Then, combining (5.1) and (5.2), we can evaluate the integral of a product of
general Wilson loops of the chiral primitive cycles at large $N$ as

$\displaystyle\Bigl{\langle}\prod_{C\in\Pi^{+}}|\Upsilon_{\lambda}(P_{C}(U))|^{2}\Bigr{\rangle}\underset{N\to\infty}{\longrightarrow}\prod_{C\in\Pi^{+}}z_{\lambda}\,,$
(5.3)

where $z_{\lambda}$ is defined by (4.10). See [28] for more detail.

We can evaluate (3.5) by using these facts. Since the matrix weighted
Bartholdi zeta function can be written as (2.16), we can evaluated the
integration in (3.5) as

$\displaystyle{\cal V}_{G}(q,u)^{-\frac{N^{2}}{2}}$
$\displaystyle\int\prod_{e\in E}dU_{e}\,\zeta_{G}(q,u;U)^{\frac{1}{2}}$
$\displaystyle=\Bigl{\langle}\prod_{C\in\Pi^{+}}e^{\sum_{m=1}^{\infty}\frac{1}{m}f_{C}(q,u)^{m}|\operatorname{Tr}P_{C}(U)^{m}|^{2}}\Bigr{\rangle}$
$\displaystyle=\Bigl{\langle}\prod_{C\in\Pi^{+}}\sum_{n=1}^{\infty}f_{C}(q,u)^{n}\sum_{\lambda\vdash
n}\frac{1}{z_{\lambda}}|\Upsilon_{\lambda}(P_{C}(U))|^{2}\Bigr{\rangle}$
$\displaystyle\underset{N\to\infty}{\longrightarrow}\prod_{C\in\Pi^{+}}\sum_{n=1}^{\infty}{f_{C}(q,u)^{n}}p(n)=\prod_{C\in\Pi^{+}}\prod_{k=1}^{\infty}\frac{1}{1-f_{C}(q,u)^{k}}\,,$
(5.4)

where we have use the large $N$ decomposition (5.3) in the final line and
$p(n)$ is the number of the partitions of $n$. Therefore, the partition
function (3.3) in the large $N$ limit behaves as

$\displaystyle Z_{\rm
gKM}\underset{N\to\infty}{\longrightarrow}\left(\frac{2\pi}{\beta}\right)^{\frac{1}{2}n_{V}N^{2}}(1-(1-u)^{2}q^{2})^{\frac{1}{2}{(n_{E}-n_{V})N^{2}}}{\cal
V}_{G}(q,u)^{\frac{N^{2}}{2}}\prod_{C\in\Pi^{+}}\prod_{k=1}^{\infty}\frac{1}{1-f_{C}(q,u)^{k}}\,.$
(5.5)

Note that, the partition function of the gKM model on a $d$-regular graph
(3.12) can be evaluated using the same technique as

$\displaystyle Z_{\rm gKM}$
$\displaystyle\underset{N\to\infty}{\longrightarrow}\left(\frac{2\pi}{\beta
q\,m^{2}}\right)^{\frac{1}{2}n_{V}N^{2}}{\cal
V}_{G}(m^{-2},1)^{\frac{N^{2}}{2}}\prod_{C\in\Pi_{+}}\prod_{k=1}^{\infty}\frac{1}{1-f_{C}(m^{-2},1)}\,.$
(5.6)

###  5.2 Regularized partition function

Since there is still $N$-dependence on the right-hand side of (5.5), the
precise meaning of this expression is that the partition function of the gKM
model is asymptotic to the right-hand side at large $N$. Therefore, it is
convenient to give a suitable approximation of the partition function at
finite $N$ which asymptotically approach the right-hand side of (5.5).

The part to consider is
$\prod_{C\in\Pi^{+}}\prod_{k=1}^{\infty}\frac{1}{1-f_{C}(q,u)^{k}}$ in the
right-hand side, which is obtained by evaluating

$\left\langle\prod_{C\in[{\cal
P}_{R}]}\sum_{n=0}^{\infty}f_{C}(q,u)^{n}\sum_{\lambda\vdash
n}\frac{1}{z_{\lambda}}|\Upsilon_{\lambda}(P_{C}(U))|^{2}\right\rangle\,.$
(5.7)

We can safely use the large $N$ decompositions (5.1) and (5.2) because all the
class functions $\Upsilon_{\lambda}(U)$ for $U\in U(N)$ are independent
regardless of the size of the partition $\lambda$. However, at finite $N$, we
have to take care of the size of the partition since
$\operatorname{Tr}(U^{n})$ for $n>N$ can be expanded by products of
$\operatorname{Tr}(U^{k})$’s ($k\leq N$)444 This is because
$\operatorname{Tr}(U^{n})$ is the $n$’s power-sum symmetric polynomial of the
$N$ eigenvalues of $U$.. We therefore restrict the partitions to those with
row lengths less than or equal to $N$ so that we count only independent Wilson
loops. If we ignore the ${\cal O}(1/N)$ contributions, we can further use the
large $N$ decomposition (5.3) as an approximation. Then we can approximate
(5.7) as

$\displaystyle\left\langle\prod_{C\in[{\cal
P}_{R}]}\sum_{n=0}^{\infty}f_{C}(q,u)^{n}\sum_{\lambda\vdash
n}\frac{1}{z_{\lambda}}|\Upsilon_{\lambda}(P_{C}(U))|^{2}\right\rangle$
$\displaystyle\simeq\prod_{C\in[{\cal
P}_{R}]}\sum_{n=0}^{\infty}f_{C}(q,u)^{n}\sum_{\underset{|\lambda_{i}|\leq
N}{\lambda\vdash
n}}\frac{1}{z_{\lambda}}\left\langle|\Upsilon_{\lambda}(P_{C}(U))|^{2}\right\rangle$
$\displaystyle=\prod_{C\in[{\cal
P}_{R}]}\sum_{n=0}^{\infty}f_{C}(q,u)^{n}p(n;N)$
$\displaystyle=\prod_{C\in[{\cal
P}_{R}]}\prod_{k=1}^{N}\frac{1}{1-f_{C}(q,u)^{k}}\,,$ (5.8)

where $p(n;N)$ is the number of Young tableau with $n$ boxes and the lengths
of the rows are less than or equal to $N$. We then propose the approximate
partition function at finite $N$ as

$\displaystyle\hat{Z}_{\rm
gKM}\equiv\left(\frac{2\pi}{\beta}\right)^{\frac{1}{2}n_{V}N^{2}}(1-(1-u)^{2}q^{2})^{\frac{1}{2}{(n_{E}-n_{V})N^{2}}}{\cal
V}_{G}(q,u)^{\frac{N^{2}}{2}}\prod_{C\in\Pi^{+}}\prod_{k=1}^{N}\frac{1}{1-f_{C}(q,u)^{k}}\,.$
(5.9)

It is obvious that (5.9) asymptotically approaches the right-hand side of
(5.5)555 The difference between the regularized free energy (5.10) and the
exact free energy at finite $N$ is ${\cal O}(1)$ because the large $N$
decomposition is justified by ignoring ${\cal O}(1/N)$ contributions from
expectation values. Therefore we can use the regularized partition function
(5.9) if we are interested in the leading and the next leading behavior in
$1/N$-expansion.  in the large $N$ limit. Furthermore, the exact partition
function of the model for $G=C_{n}$ at finite $N$ coincide to (5.9). It
supports that the expression (5.9) can be regarded as a regularized partition
function of the gKM model at finite $N$.

The free energy calculated from this regularized partition function becomes

$\displaystyle\hat{\cal F}_{\rm gKM}$
$\displaystyle\equiv-\frac{2}{n_{V}}\log\hat{Z}_{\rm gKM}$
$\displaystyle\simeq N^{2}\left(-\frac{1}{n_{V}}\log{\cal
V}_{G}(q,u)\right)+\frac{2}{n_{V}}\sum_{C\in\Pi_{+}}\sum_{k=1}^{N}\log\left(1-f_{C}(q,u)^{k}\right)\,,$
(5.10)

up to irrelevant terms. It is remarkable that the leading ${\cal O}(N^{2})$
contribution to the free energy comes from ${\cal V}_{G}(q,u)$ which counts
the collapsing cycles, which explicitly realizes the analysis given in [22,
21]. On the contrary, the second term is the contribution from non-zero area
Wilson loops but is ${\cal O}(N)$. This suggests that, if we deal only with
the leading of the 1/N expansion of this theory, we completely ignore the
physics arising from the non-zero area Wilson loop. We will see it explicitly
in the next subsection.

###  5.3 Relation to the exact solution at large $N$

For the original KM model (1.1) defined on the $D$-dimensional square lattice,
the equations satisfied by the density of eigenvalues of the scalar field for
large $N$ are derived in [5], and it was found that the semi-circle
distribution satisfies the equations for any $D$ in [6]. Let us first briefly
review this analysis.

The basic idea is to combine the equations satisfied by the IZ integral,

$I(\Phi,\Psi)\equiv\int dUe^{N\operatorname{Tr}\left(\Phi U\Psi
U^{\dagger}\right)}\,,$

with the saddle point equation satisfied by the KM model at large $N$. More
concretely, the equations are those for the matrix functions,

$\displaystyle F(\Phi)\equiv\nabla_{\Phi}\ln I(\Phi,\Psi),\quad
G_{\lambda}(\Phi)\equiv\frac{1}{\lambda-F(\Phi)-\nabla_{\Phi}}\cdot 1\,,$

with
$(\nabla_{\Phi})_{ij}\equiv\frac{1}{N}\frac{\partial}{\partial\Phi_{ji}}$. We
introducing the density function of the eigenvalues of $\Phi$,

$\rho(\mu)\equiv\frac{1}{N}\sum_{a=1}^{N}\delta(\mu-\phi_{a})\,.$

Then, assuming that all $\Phi(x)$ have the same eigenvalue distribution in the
large $N$ limit, we can show that the IZ integral satisfies the following
equations at large $N$:

$\displaystyle{\rm Re}V^{\prime}(z)$ $\displaystyle={\cal
P}\int_{-\infty}^{\infty}\frac{d\nu}{2\pi i}\log\left(\frac{z-F(\nu)-{\rm
Re}V^{\prime}(\nu)+i\pi\rho(\nu)}{z-F(\nu)-{\rm
Re}V^{\prime}(\nu)-i\pi\rho(\nu)}\right),\quad{\rm
Im}V^{\prime}(z)=-\pi\rho(z)\,,$ (5.11)

where $V^{\prime}(\lambda)\equiv\frac{\rho(\nu)}{\lambda-\nu}$ and ${\cal
P}\int$ denotes the principal integration. Furthermore, from the action (1.1),
we obtain the saddle point equation in the large $N$,

$F(z)=\frac{1}{D}\left(\frac{1}{2}U^{\prime}(z)-{\rm
Re}V^{\prime}(z)\right)=\frac{1}{D}\left(m^{2}z-{\rm
Re}V^{\prime}(z)\right)\,,$ (5.12)

where $U(\Phi)$ is the potential of the KM model and is quadratic,
$U(\Phi)=\frac{m^{2}}{2}\Phi^{2}$, in the present case and the second term
${\rm Re}V^{\prime}(z)$ comes from the variation of the Vandermonde
determinant. Substituting (5.12) to (5.11), we can eliminate the dependence of
$F$ from the equations. For more detail, see [5].

In [6], it has been shown that the semi-circle distribution,

$\rho(\nu)=\frac{1}{\pi}\sqrt{\mu-\frac{\mu^{2}\nu^{2}}{4}}\,,$ (5.13)

solves the equations (5.11) and (5.12) for any $D$ as

$\mu_{\pm}=\frac{1}{2D-1}\Bigl{(}(D-1)m^{2}\pm D\sqrt{m^{4}-8D+4}\Bigr{)}\,,$
(5.14)

where $\mu$ is a constant determined by $D$ and $m$.

We can apply the same analysis to the gKM model. To justify the assumption
that all the scalar field have the same eigenvalue distribution, we consider a
$d$-regular graph as in Sec. 3.3. Then the saddle point equation of the gKM
model becomes

$F(z)=\frac{2}{d}\left(m^{2}z-{\rm Re}V^{\prime}(z)\right)\,.$

Comparing it to the saddle point equation for the original KM model (5.12), we
see that the equations can be solved by simply replacing $D$ with $d/2$.
Therefore, we can conclude that the semi-circle distribution (5.13) with

$\mu=\mu_{\pm}=\frac{1}{2(d-1)}\Bigl{(}(d-2)m^{2}\pm
d\sqrt{m^{4}-4(d-1)}\Bigr{)}\,,$ (5.15)

is an exact solution of the gKM model on a $d$-regular graph at large $N$.
This indicates that the semi-circle distribution is a universal scalar field
behavior independent of the details of the graph.

What type of Wilson loops are involved in this semi-circle solution? As
discussed in [6], the contribution to free energy from the semi-circle
solution (5.13) is

$\displaystyle{\cal F}_{\rm
SC}=N^{2}\left(\frac{1}{2}\frac{m^{2}}{2\mu}+\frac{1}{2}\log\mu-\frac{D}{2}\sqrt{1+\frac{4}{\mu^{2}}}-1-\log\left(\frac{1}{2}+\frac{1}{2}\sqrt{1+\frac{4}{\mu^{2}}}\right)\right)\,,$

which is ${\cal O}(N^{2})$. The $1/N$-expansion of the free energy (5.10)
suggests that the semi-circle solution comes not from the non-zero area Wilson
loops but from the zero area Wilson loops. To confirm it, let us evaluate the
expectation value of the scalar fields,

$\frac{1}{Nn_{V}}\Bigl{\langle}\sum_{v\in
V}\operatorname{Tr}\Phi_{v}^{2}\Bigr{\rangle}=\frac{1}{Nn_{V}}\Bigl{\langle}\sum_{v\in
V}\sum_{a=1}^{N}\phi_{v,a}^{2}\Bigr{\rangle}\,.$ (5.16)

If the distribution of the eigenvalues is semi-circle (5.13), the expectation
value can be evaluated as

$\frac{1}{Nn_{V}}\Bigl{\langle}\sum_{v\in V}{\rm
Tr}\Phi_{v}^{2}\Bigr{\rangle}_{\rm
SC}=\frac{1}{\pi}\int_{-\frac{2}{\sqrt{\mu}}}^{\frac{2}{\sqrt{\mu}}}dxx^{2}\sqrt{\mu-\frac{\mu^{2}x^{2}}{4}}=\frac{1}{\mu}\,.$

On the other hand, from the action (3.9), the expectation value can be
evaluated as

$\displaystyle\frac{1}{Nn_{V}}\Bigl{\langle}\sum_{v\in V}{\rm
Tr}\Phi_{v}^{2}\Bigr{\rangle}_{\rm gKM}$
$\displaystyle=-\frac{2}{N^{2}n_{V}}\frac{\partial}{\partial m^{2}}\log Z_{\rm
gKM}$ $\displaystyle=-\frac{1}{n_{V}}\frac{\partial}{\partial m^{2}}\log{\cal
V}_{G}(m^{-2},1)+\frac{1}{N^{2}n_{V}}\sum_{C\in\Pi_{+}}\sum_{k=1}^{N}\frac{\partial}{\partial
m^{2}}\log\left(1-f_{C}(m^{-2},1)^{k}\right)\,,$ (5.17)

where we have used the regularized partition function and set $\beta q=N$.
Since the functions ${\cal V}_{G}(m^{-2},1)$ and $f_{C}(m^{-2},1)$ depend on
the details of the graph, we evaluate the expectation value for the cycle
graph $C_{n}$ for which these functions can be explicitly written down as

$\displaystyle V_{C_{n}}(m^{-2},1)=R(m^{2})^{\frac{N^{2}}{2}n},\quad
f_{C}(m^{-2},1)=R(m^{2})^{n}\,,$

where $R(m^{2})$ is given by (4.16). Substituting them into (5.17), we obtain

$\displaystyle\frac{1}{Nn_{V}}\Bigl{\langle}\sum_{v\in V}{\rm
Tr}\Phi_{v}^{2}\Bigr{\rangle}_{\rm gKM}$
$\displaystyle=-\frac{R^{\prime}(m^{2})}{R(m^{2})}\left(1-\frac{1}{N^{2}}\sum_{k=1}^{N}\frac{R(m^{2})^{nk}}{1-R(m^{2})^{nk}}\right)=\frac{1}{\mu}\left(1+{\cal
O}(1/N)\right)\,,$ (5.18)

where we have used
$\frac{-R^{\prime}(m^{2})}{R(m^{2})}=\frac{1}{\sqrt{m^{4}-4}}=\frac{1}{\mu}$.
This shows that the exact solution at large $N$ reflects only the zero area
Wilson loops of the graph as expected.

###  5.4 Reduction to the Ihara zeta function

As we have seen in Sect. 2, the Bartholdi zeta function reduces to the Ihara
zeta function by setting $u=0$, which counts only the reduced cycles of the
graph. In particular, the functions ${\cal V}_{G}(q,u)$ and $f_{C}(q,u)$
become ${\cal V}_{G}(q,0)=1$ and $f_{C}(q,0)=q^{|C|}$, respectively, and the
(regularized) partition function (5.9) becomes

$\displaystyle Z_{\rm gKM}^{G,N}$
$\displaystyle=\left(\frac{2\pi}{\beta}\right)^{\frac{1}{2}n_{V}N^{2}}(1-q^{2})^{\frac{1}{2}{(n_{E}-n_{V})N^{2}}}\prod_{C\in\Pi^{+}}\prod_{k=1}^{N}\frac{1}{1-q^{|C|k}}$
$\displaystyle=\left(\frac{2\pi}{\beta}\right)^{\frac{1}{2}n_{V}N^{2}}(1-q^{2})^{\frac{1}{2}{(n_{E}-n_{V})N^{2}}}\prod_{k=1}^{N}\zeta_{G}(q^{k})^{\frac{1}{2}}\,,$
(5.19)

as shown in [28]. We saw in the previous subsection that the semi-circle
distribution is an exact solution of this model at large $N$ and it arises
from the collapsing cycles. A natural question then arises as to what happens
to this solution when the collapsing cycles vanish at $u=0$.

If the value of $u$ is fixed, we cannot evaluate
$\frac{1}{Nn_{V}}\Bigl{\langle}\sum_{v\in V}{\rm
Tr}\Phi_{v}^{2}\Bigr{\rangle}$ simply by the derivative of the mass parameter,
since the coefficients of $\sum_{v\in V}\operatorname{Tr}\Phi_{v}^{2}$ and
$\sum_{e\in E}\operatorname{Tr}\Phi_{s(e)}U_{e}\Phi_{t(e)}U_{e}^{-1}$ are
determined only by the single parameter $q$. Instead, we can use the identity
satisfied by a $d$-regular graph,

$\left(q\frac{\partial}{\partial
q}-\beta\frac{\partial}{\partial\beta}\right)\left(\frac{1}{N^{2}}\log Z_{\rm
gKM}\right)=\frac{1}{2}\left(q^{-1}-((1-u)d-(1-u)^{2})q)\right)\frac{\beta
q}{N^{2}}\Bigl{\langle}\sum_{v\in
V}\operatorname{Tr}\Phi_{v}^{2}\Bigr{\rangle}\,.$ (5.20)

In order to compare to the result (5.18) in the previous subsection, we
consider the case of $u=0$ for $d=2$. In this parametrization, $\log Z_{\rm
gKM}$ is written as

$\frac{1}{N^{2}}\log Z_{\rm
gKM}^{C_{n}}=\frac{n}{2}\log\frac{2\pi}{\beta}+{\cal O}(1/N)\,.$

where the ${\cal O}(1/N)$ part is independent of $\beta$. Since the left-hand
side of (5.20) becomes

$q\frac{\partial}{\partial q}\left(\frac{1}{N^{2}}\log Z_{\rm
gKM}^{C_{n}}\right)={\cal
O}(1/N),\quad-\beta\frac{\partial}{\partial\beta}\left(\frac{1}{N^{2}}\log
Z_{\rm gKM}^{C_{n}}\right)=\frac{n}{2}\,.$

we can evaluate the expectation value of the scalar fields as

$\frac{1}{Nn_{V}}\Bigl{\langle}\sum_{v\in V}{\rm
Tr}\Phi_{v}^{2}\Bigr{\rangle}_{\rm gKM}=\frac{1}{q^{-1}-q}+{\cal
O}(1/N)=\frac{1}{\mu}+{\cal O}(1/N)\,,$ (5.21)

where we have set $\beta q=N$ after the computation and used
$\mu=\sqrt{m^{4}-4}=q^{-1}-q$ ($|q|<1$) for $u=0$. Therefore, although we have
seen only the second momentum of the eigenvalues, it is natural to think that
the semi-circle distribution is still a solution of this model even if $u=0$.

However, this solution is of course nothing to do with the Wilson loops since
the Ihara zeta function does not count the collapsing cycles;
$V_{G}(q,u=0)=1$. The reason why the semi-circle is still a solution for $u=0$
is because the leading part of the free energy includes the contribution of
the Gaussian integral. In fact, the radius of the semi-circle distribution
obtained for $d=2$ is the same as the radius of the semi-circle distribution
produced in the large $N$ limit of the Gaussian model, obtained by simply
neglecting the contribution of the link variable from the action of the gKM
model. When $d$ is greater than 2, the semi-circle is modified by the
additional term proportional to $(n_{E}-n_{V})\log(1-(1-u)^{2}q^{2})$ from
that of the the Gaussian model. However, it is still true that no Wilson loop
contributes to the solution for $u=0$.

##  6 Conclusion and Discussion

In this paper, we proposed an extension of the Bartholdi zeta function by
putting matrices as weights on the edges of the graph and gave its determinant
representation. We showed that the partition function of the Kazakov-Migdal
model defined on a graph (gKM model) is represented by the unitary matrix
integral of the matrix weighted Bartholdi zeta function in general. We also
derived the dual expression of the partition function by integrating the
unitary matrices first using the so-called HCIZ integration formula. We
computed the partition function of the gKM model for the cycle graph $C_{n}$
exactly at finite $N$, and showed that the partition function is closely
related to the generalized Catalan numbers. As a byproduct, we found a
relation between the generating function of the generalized Catalan numbers
and that of the usual Catalan numbers. We also evaluated the partition
function of the gKM model for arbitrary graphs at large $N$ and showed that it
can be expressed as an infinite product of deformed Ihara zeta functions. We
showed explicitly that the leading terms in the $1/N$-expansion of the free
energy are the contribution of zero area Wilson loop, while the contribution
of non-zero area Wilson loops is of the order of ${\cal O}(1/N)$ compared to
the leading. We applied the large $N$ analysis performed on the original KM
model to the gKM model and found that the semi-circle distribution is an exact
solution for the gKM model on the regular graphs in general. The contribution
from the zero area Wilson loops disappears if we set $u=0$, where the gKM
model is expressed by the Ihara zeta function.

The partition function of the gKM model is represented by the unitary matrix
integral (3.5), while at the same time having the dual expression (3.7). It is
not surprising that the dual expression (3.7) contains the sum of the symmetry
groups because unitary matrix integrals can be written in terms of symmetry
groups in general as a result of the Schur-Weyl duality. However, it is
interesting that the resulting expression includes the adjacency matrix (3.8)
of the covering graph. This strongly suggests that the Bartholdi zeta function
weighted by the unitary matrix is closely related to the Bartholdi zeta
function of the covering graph. As mentioned below (3.8), a covering graph is
generated by assigning a permutation on each edge of the graph $G$. More
generally, we can assume that an element of the finite group $\Gamma$ is
assigned on the edge $e$ of the graph as $\alpha(e)\in\Gamma$. When
$\alpha(e^{-1})=\alpha(e)^{-1}$, this is called the ordinary voltage
assignment in graph theory. Since any finite group can be represented by a
permutation group, we can easily see that the ordinary voltage assignment
$\alpha$ induces a covering graph $G^{\alpha}$ in the same way. The Bartholdi
zeta function of $G^{\alpha}$ is closely related to the base graph $G$ as
expected. It is achieved by defining the Bartholdi $L$-function [36],

$\zeta_{G}(q,u,\alpha,\rho)\equiv\prod_{C\in[{\cal
P}]}\det\left(\boldsymbol{1}-q^{|C|}u^{b(C)}\rho(\alpha(C))\right)^{-1}\,,$
(6.1)

where $\alpha(C)$ is defined as
$\alpha(C)\equiv\alpha({\boldsymbol{e}}_{1})\cdots\alpha({\boldsymbol{e}}_{r})\in\Gamma$
when the cycle is written as
$C={\boldsymbol{e}}_{1}\cdots{\boldsymbol{e}}_{r}$. As shown in [36], the
Bartholdi zeta function of $G^{\alpha}$ can be expressed through the
$L$-function as

$\zeta_{G^{\alpha}}(q,u)=\prod_{\rho}\zeta_{G}(q,u,\alpha,\rho)^{\dim\rho}\,,$

where $\dim\rho$ is the dimension of the representation $\rho$. The similarity
between the matrix weighted Bartholdi zeta function (2.12) and the Bartholdi
$L$-function (6.1) is obvious: The matrix weighted Bartholdi zeta function
would be regarded as an $L$-function when the voltage assignment to the graph
is extended to a Lie group. It is plausible to think that the appearance of
the adjacency matrix (3.8) of the covering graph will be a consequence of this
similarity. A more comprehensive understanding of the graph zeta function from
this perspective would be a promising topic for the future.

In this paper, we have restricted the parameters of the gKM model as (3.2) in
order to reveal the relation between the gKM model and the Bartholdi zeta
function, but it is not a general parametrization. The most general
parametrization is to assign an independent coupling constant to all vertices
and all edges as

$\displaystyle S_{gKM}=\operatorname{Tr}\left\\{\frac{1}{2}\sum_{v\in
V}m_{v}^{2}\Phi_{v}^{2}-\sum_{e\in
E}q_{e}\Phi_{s(e)}U_{e}\Phi_{t(e)}U_{e}^{-1}\right\\}\,.$ (6.2)

Using these degrees of freedom, we will be able to see a connection to a more
general graph zeta function than the Bartholdi zeta function. For example, the
Bartholdi zeta function is generalized in [45] where a different value of the
bump parameter $u_{v}$ is assigned to every vertex $v\in V$ as

$\zeta_{G}(q,u_{1},\cdots,u_{n_{V}})\equiv\prod_{C\in{\cal
P}}\left(1-q^{|C|}\prod_{v\in V}u_{v}^{b_{v}(C)}\right)^{-1}\,,$ (6.3)

where $b_{v}(C)$ is the number of bumps $e_{i}=e_{i+1}^{-1}$ with the
condition $t(e_{i})=v$. We can reproduce (the matrix weighted version of) this
zeta function by tuning the parameters of (6.2) as

$m_{v}^{2}=1+q^{2}\sum_{v^{\prime}:\text{neighbor of
}v}\frac{1-u_{v^{\prime}}}{1-q^{2}(1-u_{v})(1-u_{v^{\prime}})},\quad
q_{e}=\frac{q}{1-q^{2}(1-u_{s(e)})(1-u_{t(e)})}\,,$ (6.4)

as the partition function up to an overall factor. It would be one of the
future problems to examine the nature of the gKM model and compare it to the
generalized Bartholdi zeta function. The more interesting question, however,
is for what category of parameters the partition function of the gKM model has
an Euler product representation. Although the zeta function is generally
defined through an Euler product, it is not obvious whether it has a
determinant representation. Conversely, it is clear that the partition
function of the gKM model can be expressed in determinant form, but it is not
obvious whether it has a meaning as a zeta function or not. The Ihara and
Bartholdi zeta functions are interesting zeta functions that have both Euler
product and determinant representations, and the fact that they are associated
with the gKM model suggests the possibility of shedding light on more general
properties of the zeta function through this model. It is quite an interesting
future problem to pursue this possibility.

We finally would like to make a comment on another possibility of the gKM
model. The main part of the $1/N$ expansion of the gKM model reflects only
zero area Wilson loops and the contribution of non-zero area Wilson loops,
which is of most interest for gauge theory, appears rather in the sub-leading
part of the $1/N$ expansion. If we adjust the theory so that the contribution
of the leading part is appropriately eliminated, we will be able to construct
a sort of gauge theory in the continuum limit. It is of course not the usual
QCD because of the existence of the center gauge symmetry, but it is still a
summation of non-zero area Wilson loops. In that case, the existence of the
parameter $u$ is expected to cause interesting phenomena: From the Wilson
loop’s point of view, allowing bumps corresponds to considering the zigzag
symmetry of the Wilson loop, which is essential for gauge theories to behave
like string theory in the large $N$ limit [46]. It would be an interesting
attempt in the future to use the graph zeta function to gain insight into the
holographic principle.

## Acknowledgments

The authors would like to thank S. Aoki, S. Koyama, M. Fukuma, Y. Furuta, T.
Misumi, S. Nishigaki, H. Watanabe, Y. Yoshida and J. Yumoto for useful
discussions. This work is supported in part by Grant-in-Aid for Scientific
Research (KAKENHI) (C), Grant Number 20K03934 (S. M.).

##  Appendix A A proof of Amitsur’s theorem

In this appendix, we give a proof of Amitsur’s theorem in order to make the
paper self-contained.

Let us consider words generated by the “letters” $\\{1,\cdots,k\\}$, that is,
sequences of the $n$ letters. We call the number of the letters in a word the
length of the word, and call a word primitive (or indecomposable) if it cannot
be written as $w^{n}$ $(n\geq 2)$ for a word $w$ with a shorter length. We
call the words $w_{1}$ and $w_{2}$ are equivalent when $w_{1}$ is obtained by
a cyclic rotation of the letters in $w_{2}$. In the following, We denote the
set of primitive words of length $d$ as $P_{d}$, the set of the representative
of the primitive words of length $d$ as $L_{d}$, and the set of the
representative of all primitive words as $L\equiv\bigcup_{d=1}^{\infty}L_{d}$.

Let us consider $n$ matrices $\\{X_{1},\cdots,X_{k}\\}$. Corresponding to a
word $w=i_{1}\cdots i_{n}$, we define

$X_{w}\equiv X_{i_{1}}\cdots X_{i_{n}}\,.$

Then, it is obvious that the quantity $(X_{1}+\cdots+X_{k})^{m}$ is a
summation of $X_{w}$ where $w$ runs all words of length $m$. Recalling that
any word can be expressed as a positive power of a primitive word, we can
write this quantity as

$(X_{1}+\cdots+X_{k})^{m}=\sum_{d|m}\sum_{\tilde{w}\in
P_{d}}X_{\tilde{w}}^{m/d}\,.$ (A.1)

We here note that, if $w$ is a primitive word of length $d$, there are $d$
different equivalent words to $w^{n}$ ($n\geq 1$). Therefore, we can write the
trace of the expression (A.1) as

$\operatorname{Tr}(X_{1}+\cdots+X_{k})^{m}=\sum_{d|m}d\sum_{w\in
L_{d}}\operatorname{Tr}(X_{w}^{m/d})\,.$

Using this result, we can show the identity,

$\displaystyle\log\det\left(\boldsymbol{1}-X_{1}-\cdots-X_{k}\right)^{-1}$
$\displaystyle=\sum_{m=1}^{\infty}\frac{1}{m}\operatorname{Tr}\left(X_{1}+\cdots+X_{k}\right)^{m}$
$\displaystyle=\sum_{m=1}^{\infty}\frac{1}{m}\sum_{d|m}d\sum_{w\in
L_{d}}\operatorname{Tr}(X_{w}^{m/d})$
$\displaystyle=\sum_{k=1}^{\infty}\frac{1}{k}\sum_{d=1}^{\infty}\sum_{w\in
L_{d}}\operatorname{Tr}(X_{w}^{k})$ $\displaystyle=\log\prod_{w\in
L}\det\left(\boldsymbol{1}-X_{w}\right)^{-1}\,.$

We then obtain

$\det\left(\boldsymbol{1}-X_{1}-\cdots-X_{k}\right)=\prod_{w\in
L}\det\left(\boldsymbol{1}-X_{w}\right)\,.$

##  Appendix B Proof of the identity (2.15)

In this appendix, we show the identity (2.15), that is,

$f_{C,n}(q,u)=f_{C}(q,u)^{n}\,$ (B.1)

where

$\begin{split}f_{C,n}(q,u)&\equiv\sum_{k|n}k\sum_{{\tilde{C}}\in[{\cal
B}(C^{k})]}(q^{|{\tilde{C}}|}u^{b({\tilde{C}})})^{n/k}\,,\\\
f_{C}(q,u)&\equiv\sum_{{\tilde{C}}\in[{\cal
B}(C)]}q^{|{\tilde{C}}|}u^{b({\tilde{C}})}\,,\end{split}$

for a representative of primitive cycles $C$.

We first note that, if we label the elements of $[{\cal B}(C)]$, that is, the
set of cycles that reduce to $C$, as $C_{j}\in[{\cal B}(C)]$
($j\in\mathbb{N})$, ${}^{\forall}{\tilde{C}}\in[{\cal B}(C^{d})]$ can be
uniquely written as

${\tilde{C}}=C^{\prime}_{j_{1}}\cdots C^{\prime}_{j_{d}}\,,$

where each $C^{\prime}_{j_{i}}$ is an equivalent cycle to $C_{j_{i}}$. This
means that, as far as we consider such quantity that is irrelevant to the
cyclic rotation of the cycles in $[{\cal B}(C)]$, we can identify an element
of $[{\cal B}(C^{d})]$ as a word of length $d$ generated by the elements of
$[{\cal B}(C)]$. Therefore, we rather consider words generated by $[{\cal
B}(C)]$, instead of cycles, in the following.

We then consider the summation of all possible words with length $n$ made of
the elements of $[{\cal B}(C)]$, which is given by
$\left(\sum_{{\tilde{C}}\in[{\cal B}(C)]}{\tilde{C}}\right)^{n}$. Since any
word can be expressed as a positive power of a primitive word in general, we
can describe this quantity as

$\left(\sum_{{\tilde{C}}\in[{\cal
B}(C)]}{\tilde{C}}\right)^{n}=\sum_{k|n}\sum_{{\tilde{C}}\in
P_{k}}{\tilde{C}}^{\,n/k}\,,$ (B.2)

where $P_{k}$ denotes the set of primitive words of length $k$. Since an
element of $[{\cal B}(C^{k})]$ is originally a set of representatives of
primitive cycles, it is also primitive in the sense of words. Therefore there
are $d$ different words which are equivalent to ${\tilde{C}}\in[{\cal
B}(C^{k})]$. If we define a mapping $t(C)$ by

$t(C)=q^{|C|}u^{b(C)}\,,$

it is obvious that $t(C)=t(C^{\prime})$ if $C\sim C^{\prime}$. So we do not
need to care about the differences between words and cycles. Therefore, by
acting this mapping to both side of (B.2), we finally obtain

$\left(\sum_{{\tilde{C}}\in[{\cal
B}(C)]}q^{|{\tilde{C}}|}u^{b({\tilde{C}})}\right)^{n}=\sum_{k|n}k\sum_{{\tilde{C}}\in[{\cal
B}(C^{k})]}(q^{|{\tilde{C}}|}u^{b({\tilde{C}})})^{n/k}\,,$

which is nothing but the equality (B.1).

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