Ronkin/Zeta Correspondence

First we introduce the following notation: $\mathbb{Z}$ is the set of integers, $\mathbb{Z}_{\geq}$ is the set of non-negative integers, $\mathbb{Z}_{>}$ is the set of positive integers, $\mathbb{R}$ is the set of real numbers, and $\mathbb{C}$ is the set of complex numbers. Moreover, $T^{d}_{N}$ denotes the $d$-dimensional torus with $N^{d}$ vertices, where $d,\ N\in\mathbb{Z}_{>}$. Remark that $T^{d}_{N}=(\mathbb{Z}\ \mbox{mod}\ N)^{d}$.

Following [8] in which Walk/Zeta Correspondence on $T^{d}_{N}$ was investigated, we treat our setting for $2d$-state discrete time walk with a nearest-neighbor jump on $T^{d}_{N}$.

The discrete time walk is defined by using a shift operator and a coin matrix which will be mentioned below. Let $f:T^{d}_{N}\longrightarrow\mathbb{C}^{2d}$. For $j=1,2,\ldots,d$ and $\bm{x}\in T^{d}_{N}$, the shift operator $\tau_{j}$ is defined by $(\tau_{j}f)(\bm{x})=f(\bm{x}-\bm{e}_{j})$, where $\{\bm{e}_{1},\bm{e}_{2},\ldots,\bm{e}_{d}\}$ denotes the standard basis of $T^{d}_{N}$. This means that the walks are taken along the direction $\{\pm\bm{e}_{1},\pm\bm{e}_{2},\ldots,\pm\bm{e}_{d}\}$. Therefore, the direction polytope ${\rm DP}_{d}$ of degree $d$ can be defined as follows:

$${\rm DP}_{d}={\rm Conv}(\pm\bm{e}_{1},\pm\bm{e}_{2},\ldots,\pm\bm{e}_{d}),$$

where the symbol ${\rm Conv}(S)$ means the convex hull spanned by a subset $S\subset T^{d}_{N}$. Moreover, we define the direction graph ${\rm DG}_{d}$ of degree $d$ as the dual graph of ${\rm DP}_{d}$.

The coin matrix $A=[a_{ij}]_{i,j=1,2,\ldots,2d}$ is a $2d\times 2d$ matrix with $a_{ij}\in\mathbb{C}$ for $i,j=1,2,\ldots,2d$. If $a_{ij}\in[0,1]$ and $\sum_{i=1}^{2d}a_{ij}=1$ for any $j=1,2,\ldots,2d$, then the walk is a CRW. We should remark that, in particular, when $a_{i1}=a_{i2}=\cdots=a_{i2d}$ for any $i=1,2,\ldots,2d$, this CRW becomes a RW. If $A$ is unitary, then the walk is a QW. So our class of walks contains RW, CRW, and QW as special models.

To describe the evolution of the walk, we decompose the $2d\times 2d$ coin matrix $A$ as

$\displaystyle A=\sum_{j=1}^{2d}P_{j}A,$

where $P_{j}$ denotes the orthogonal projection onto the one-dimensional subspace $\mathbb{C}\eta_{j}$ in $\mathbb{C}^{2d}$. Here $\{\eta_{1},\eta_{2},\ldots,\eta_{2d}\}$ denotes a standard basis on $\mathbb{C}^{2d}$.

The discrete time walk associated with the coin matrix $A$ on $T^{d}_{N}$ is determined by the $2dN^{d}\times 2dN^{d}$ matrix

$\displaystyle M_{A}=\sum_{j=1}^{d}\Big{(}P_{2j-1}A\tau_{j}^{-1}+P_{2j}A\tau_{j}\Big{)}.$

(1)

The state at time $n\in\mathbb{Z}_{\geq}$ and location $\bm{x}\in T^{d}_{N}$ can be expressed by a $2d$-dimensional vector:

$\displaystyle\Psi_{n}(\bm{x})={}^{T}\left[\Psi^{1}_{n}(\bm{x}),\Psi^{2}_{n}(\bm{x}),\ldots,\Psi^{2d}_{n}(\bm{x})\right]\in\mathbb{C}^{2d},$

where $T$ is the transposed operator. For $\Psi_{n}:T^{d}_{N}\longrightarrow\mathbb{C}^{2d}\ (n\in\mathbb{Z}_{\geq})$, Eq. (1) gives the evolution of the walk as follows.

$\displaystyle\Psi_{n+1}(\bm{x})\equiv(M_{A}\Psi_{n})(\bm{x})=\sum_{j=1}^{d}\Big{(}P_{2j-1}A\Psi_{n}(\bm{x}+\bm{e}_{j})+P_{2j}A\Psi_{n}(\bm{x}-\bm{e}_{j})\Big{)}.$

(2)

This equation means that the walker moves at each step one unit to the $-x_{j}$-axis direction with matrix $P_{2j-1}A$ or one unit to the $x_{j}$-axis direction with matrix $P_{2j}A$ for $j=1,2,\ldots,d$.

Moreover, for $n\in\mathbb{Z}_{>}$ and $\bm{x}=(x_{1},x_{2},\ldots,x_{d})\in T^{d}_{N}$, the $2d\times 2d$ matrix $\Phi_{n}(x_{1},x_{2},\ldots,x_{d})$ is given by

$\displaystyle\Phi_{n}(x_{1},x_{2},\ldots,x_{d})=\sum_{\ast}\Xi_{n}\left(l_{1},l_{2},\ldots,l_{2d-1},l_{2d}\right),$

where the $2d\times 2d$ matrix $\Xi_{n}\left(l_{1},l_{2},\ldots,l_{2d-1},l_{2d}\right)$ is the sum of all possible paths in the trajectory of $l_{2j-1}$ steps $-x_{j}$-axis direction and $l_{2j}$ steps $x_{j}$-axis direction and $\sum_{\ast}$ is the summation over $\left(l_{1},l_{2},\ldots,l_{2d-1},l_{2d}\right)\in(\mathbb{Z}_{\geq})^{2d}$ satisfying

$\displaystyle l_{1}+l_{2}+\cdots+l_{2d-1}+l_{2d}=n,\qquad x_{j}=-l_{2j-1}+l_{2j}\quad(j=1,2,\ldots,d).$

Here we put

$\displaystyle\Phi_{0}(x_{1},x_{2},\ldots,x_{d})=\left\{\begin{array}[]{ll}I_{2d}&\mbox{if $(x_{1},x_{2},\ldots,x_{d})=(0,0,\ldots,0)$, }\\ O_{2d}&\mbox{if $(x_{1},x_{2},\ldots,x_{d})\not=(0,0,\ldots,0)$},\end{array}\right.$

where $I_{n}$ is the $n\times n$ identity matrix and $O_{n}$ is the $n\times n$ zero matrix. Then, for the walk starting from $(0,0,\ldots,0)$, we obtain

$\displaystyle\Psi_{n}(x_{1},x_{2},\ldots,x_{d})=\Phi_{n}(x_{1},x_{2},\ldots,x_{d})\Psi_{0}(0,0,\ldots,0)\qquad(n\in\mathbb{Z}_{\geq}).$

We call $\Phi_{n}(\bm{x})=\Phi_{n}(x_{1},x_{2},\ldots,x_{d})$ matrix weight at time $n\in\mathbb{Z}_{\geq}$ and location $\bm{x}\in T^{d}_{N}$ starting from ${\bf 0}=(0,0,\ldots,0)$. When we consider the walk on not $T^{d}_{N}$ but $\mathbb{Z}^{d}$, we add the superscript “$(\infty)$” to the notation like $\Psi^{(\infty)}$ and $\Xi^{(\infty)}$.

This type is moving shift model called M-type here. Another type is flip-flop shift model called F-type whose coin matrix is given by

$\displaystyle A^{(f)}=\left(I_{d}\otimes\sigma\right)A,$

where $\otimes$ is the tensor product and

$\displaystyle\sigma=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}.$

The F-type model is also important, since it has a central role in the Konno-Sato theorem [16], for example. When we distinguish $A$ (M-type) from $A^{(f)}$ (F-type), we write $A$ by $A^{(m)}$.

The measure $\mu_{n}(\bm{x})$ at time $n\in\mathbb{Z}_{\geq}$ and location $\bm{x}\in T^{d}_{N}$ is defined by

$\displaystyle\mu_{n}(\bm{x})=\|\Psi_{n}(\bm{x})\|_{\mathbb{C}^{2d}}^{p}=\sum_{j=1}^{2d}|\Psi_{n}^{j}(\bm{x})|^{p},$

where $\|\cdot\|_{\mathbb{C}^{2d}}^{p}$ denotes the standard $p$-norm on $\mathbb{C}^{2d}$. As for RW and QW, we take $p=1$ and $p=2$, respectively. Then RW and QW satisfy

$\displaystyle\sum_{\bm{x}\in T_{N}^{d}}\mu_{n}(\bm{x})=\sum_{\bm{x}\in T_{N}^{d}}\mu_{0}(\bm{x}),$

for any time $n\in\mathbb{Z}_{>}$.

To consider the zeta function, we use the Fourier analysis. To do so, we introduce the following notation: $\mathbb{K}_{N}=\{0,1,\ldots,N-1\}$ and $\widetilde{\mathbb{K}}_{N}=\{0,2\pi/N,\ldots,2\pi(N-1)/N\}$.

For $f:\mathbb{K}_{N}^{d}\longrightarrow\mathbb{C}^{2d}$, the Fourier transform of the function $f$, denoted by $\widehat{f}$, is defined by the sum

$\displaystyle\widehat{f}(\bm{k})=\frac{1}{N^{d/2}}\sum_{\bm{x}\in\mathbb{K}_{N}^{d}}e^{-2\pi i\langle\bm{x},\bm{k}\rangle/N}\ f(\bm{x}),$

(3)

where $\bm{k}=(k_{1},k_{2},\ldots,k_{d})\in\mathbb{K}_{N}^{d}$. Here $\langle\bm{x},\bm{k}\rangle$ is the canonical inner product of $\mathbb{R}^{d}$, i.e., $\langle\bm{x},\bm{k}\rangle=\sum_{j=1}^{d}x_{j}k_{j}$. Then we see that $\widehat{f}:\mathbb{K}_{N}^{d}\longrightarrow\mathbb{C}^{2d}$. Moreover, we should remark that

$\displaystyle f(\bm{x})=\frac{1}{N^{d/2}}\sum_{\bm{k}\in\mathbb{K}_{N}^{d}}e^{2\pi i\langle\bm{x},\bm{k}\rangle/N}\ \widehat{f}(\bm{k}),$

(4)

where $\bm{x}=(x_{1},x_{2},\ldots,x_{d})\in\mathbb{K}_{N}^{d}$. By using

$\displaystyle\widetilde{k}_{j}=\frac{2\pi k_{j}}{N}\in\widetilde{\mathbb{K}}_{N},\quad\widetilde{\bm{k}}=(\widetilde{k}_{1},\widetilde{k}_{2},\ldots,\widetilde{k}_{d})\in\widetilde{\mathbb{K}}_{N}^{d},$

(5)

we can rewrite Eqs. (3) and (4) in the following way:

$\displaystyle\widehat{g}(\widetilde{\bm{k}})=\frac{1}{N^{d/2}}\sum_{\bm{x}\in\mathbb{K}_{N}^{d}}e^{-i\langle\bm{x},\widetilde{\bm{k}}\rangle}\ g(\bm{x}),\qquad g(\bm{x})=\frac{1}{N^{d/2}}\sum_{\widetilde{\bm{k}}\in\widetilde{\mathbb{K}}_{N}^{d}}e^{i\langle\bm{x},\widetilde{\bm{k}}\rangle}\ \widehat{g}(\widetilde{\bm{k}}),$

for $g:\mathbb{K}_{N}^{d}\longrightarrow\mathbb{C}^{2d}$ and $\widehat{g}:\widetilde{\mathbb{K}}_{N}^{d}\longrightarrow\mathbb{C}^{2d}$. In order to take a limit $N\to\infty$, we introduced the notation given in Eq. (5). We should note that as for the summation, we sometimes write “$\bm{k}\in\mathbb{K}_{N}^{d}$” instead of “$\widetilde{\bm{k}}\in\widetilde{\mathbb{K}}_{N}^{d}$”. From the Fourier transform and Eq. (5), we have

$\displaystyle\widehat{\Psi}_{n+1}(\bm{k})=\widehat{M}_{A}(\bm{k})\widehat{\Psi}_{n}(\bm{k}),$

where $\Psi_{n}:T_{N}^{d}\longrightarrow\mathbb{C}^{2d}$ and $2d\times 2d$ matrix $\widehat{M}_{A}(\bm{k})$ is determined by

$\displaystyle\widehat{M}_{A}(\bm{k})=\sum_{j=1}^{d}\Big{(}e^{2\pi ik_{j}/N}P_{2j-1}A+e^{-2\pi ik_{j}/N}P_{2j}A\Big{)}.$

By using notations in Eq. (5), we get

$\displaystyle\widehat{M}_{A}(\widetilde{\bm{k}})=\sum_{j=1}^{d}\Big{(}e^{i\widetilde{k}_{j}}P_{2j-1}A+e^{-i\widetilde{k}_{j}}P_{2j}A\Big{)}.$

(6)

Next we will consider the following eigenvalue problem for $2dN^{d}\times 2dN^{d}$ matrix $M_{A}$:

$\displaystyle\lambda\Psi=M_{A}\Psi,$

(7)

where $\lambda\in\mathbb{C}$ is an eigenvalue and $\Psi(\in\mathbb{C}^{2dN^{d}})$ is the corresponding eigenvector. Noting that Eq. (7) is closely related to Eq. (2), we see that Eq. (7) is rewritten as

$\displaystyle\lambda\Psi(\bm{x})=(M_{A}\Psi)(\bm{x})=\sum_{j=1}^{d}\Big{(}P_{2j-1}A\Psi(\bm{x}+\bm{e}_{j})+P_{2j}A\Psi(\bm{x}-\bm{e}_{j})\Big{)},$

(8)

for any $\bm{x}\in\mathbb{K}_{N}^{d}$. From the Fourier transform and Eq. (7), we obtain

$\displaystyle\lambda\widehat{\Psi}(\bm{k})=\widehat{M}_{A}(\bm{k})\widehat{\Psi}(\bm{k}),$

for any $\bm{k}\in\mathbb{K}_{N}^{d}$. Then the characteristic polynomials of $2d\times 2d$ matrix $\widehat{M}_{A}(\bm{k})$ for fixed $\bm{k}(\in\mathbb{K}_{N}^{d})$ is

$\displaystyle\det\Big{(}\lambda I_{2d}-\widehat{M}_{A}(\bm{k})\Big{)}=\prod_{j=1}^{2d}\Big{(}\lambda-\lambda_{j}(\bm{k})\Big{)},$

(9)

where $\lambda_{j}(\bm{k})$ are eigenvalues of $\widehat{M}_{A}(\bm{k})$. Similarly, the characteristic polynomials of $2dN^{d}\times 2dN^{d}$ matrix $\widehat{M}_{A}$ is

$\displaystyle\det\Big{(}\lambda I_{2dN^{d}}-\widehat{M}_{A}\Big{)}=\prod_{j=1}^{2d}\prod_{\bm{k}\in\mathbb{K}_{N}^{d}}\Big{(}\lambda-\lambda_{j}(\bm{k})\Big{)}.$

Therefore, by taking $\lambda=1/u$, we get the following key result.

$\displaystyle\det\Big{(}I_{2dN^{d}}-uM_{A}\Big{)}=\det\Big{(}I_{2dN^{d}}-u\widehat{M}_{A}\Big{)}=\prod_{j=1}^{2d}\prod_{\bm{k}\in\mathbb{K}_{N}^{d}}\Big{(}1-u\lambda_{j}(\bm{k})\Big{)}.$

(10)

We should note that for fixed $\bm{k}(\in\mathbb{K}_{N}^{d})$, eigenvalues of $2d\times 2d$ matrix $\widehat{M}_{A}(\bm{k})$ are expressed as

$\displaystyle{\rm Spec}(\widehat{M}_{A}(\bm{k}))=\left\{\lambda_{j}(\bm{k})\ |\ j=1,2,\ldots,2d\right\}.$

Moreover, eigenvalues of $2dN^{d}\times 2dN^{d}$ matrix not only $\widehat{M}_{A}$ but also $M_{A}$ are expressed as

$\displaystyle{\rm Spec}(\widehat{M}_{A})={\rm Spec}(M_{A})=\left\{\lambda_{j}(\bm{k})\ |\ j=1,2,\ldots,2d,\ \bm{k}\in\mathbb{K}_{N}^{d}\right\}.$

By using notations in Eq. (5) and Eq. (9), we see that for fixed $\bm{k}(\in\mathbb{K}_{N}^{d})$,

$\displaystyle\det\Big{(}I_{2d}-u\widehat{M}_{A}(\widetilde{\bm{k}})\Big{)}=\prod_{j=1}^{2d}\Big{(}1-u\lambda_{j}(\widetilde{\bm{k}})\Big{)}.$

(11)

Furthermore, Eq. (6) gives the following important formula.

$\displaystyle\det\Big{(}I_{2d}-u\widehat{M}_{A}(\widetilde{\bm{k}})\Big{)}=\det\left(I_{2d}-u\times\sum_{j=1}^{d}\Big{(}e^{i\widetilde{k}_{j}}P_{2j-1}A+e^{-i\widetilde{k}_{j}}P_{2j}A\Big{)}\right).$

In this setting, we define the walk-type zeta function by

$\displaystyle\overline{\zeta}\left(A,T^{d}_{N},u\right)=\det\Big{(}I_{2dN^{d}}-uM_{A}\Big{)}^{-1/N^{d}}.$

(12)

We should remark that our walk is defined on the “site” $\bm{x}(\in T^{d}_{N})$. On the other hand, the walk in [7] is defined on the “arc” (i.e., oriented edge). However, both of the walks are the same for the torus case. As for a more detailed information, see Remark 1 in [5], for instance.

By Eqs. (10), (11) and (12), we get

$\displaystyle\overline{\zeta}\left(A,T^{d}_{N},u\right)^{-1}=\exp\left[\frac{1}{N^{d}}\sum_{\widetilde{\bm{k}}\in\widetilde{\mathbb{K}}_{N}^{d}}\log\left\{\det\Big{(}I_{2d}-u\widehat{M}_{A}(\widetilde{\bm{k}})\Big{)}\right\}\right].$

Sometimes we write $\sum_{\bm{k}\in\mathbb{K}_{N}^{d}}$ instead of $\sum_{\widetilde{\bm{k}}\in\widetilde{\mathbb{K}}_{N}^{d}}$. Noting $\widetilde{k_{j}}=2\pi k_{j}/N\ (j=1,2,\ldots,d)$ and taking a limit as $N\to\infty$, we show

$\displaystyle\lim_{N\to\infty}\overline{\zeta}\left(A,T^{d}_{N},u\right)^{-1}=\exp\left[\int_{[0,2\pi)^{d}}\log\left\{\det\Big{(}I_{2d}-u\widehat{M}_{A}\left(\Theta^{(d)}\right)\Big{)}\right\}d\Theta^{(d)}_{unif}\right],$

if the limit exists for a suitable range of $u\in\mathbb{R}$. We should note that when we take a limit as $N\to\infty$, we assume that the limit exists throughout this paper. Here $\Theta^{(d)}=(\theta_{1},\theta_{2},\ldots,\theta_{d})(\in[0,2\pi)^{d})$ and $d\Theta^{(d)}_{unif}$ denotes the uniform measure on $[0,2\pi)^{d}$, that is,

$\displaystyle d\Theta^{(d)}_{unif}=\frac{d\theta_{1}}{2\pi}\cdots\frac{d\theta_{d}}{2\pi}.$

Then the following result was obtained.

We define the logarithmic zeta function on a finite torus, and state its properties. Let $T^{d}_{N}$ denotes the $d$-dimensional torus with $N^{d}$ vertices, and a $2d\times 2d$ matrix $A=[a_{ij}]_{i,j=1,2,\ldots,2d}$ the coin matrix of a dicrete time walk on $T^{d}_{N}$, where $a_{ij}\in\mathbb{C}$ for $i,j=1,2,\ldots,2d$.

Now, we introduce the logarithmic zeta function as follows(see [13]).

$\displaystyle{\cal L}\left(A,T_{\infty}^{d},u\right)=\log\left[\lim_{N\to\infty}\left\{\overline{\zeta}\left(A,T_{N}^{d},u\right)^{-1}\right\}\right].$

(13)

Then, the second equation in Theorem 1 immediately gives

Moreover, we define $C_{r}(A,T^{d}_{N})$ by

$\displaystyle\overline{\zeta}\left(A,T^{d}_{N},u\right)=\exp\left(\sum_{r=1}^{\infty}\frac{C_{r}(A,T^{d}_{N})}{r}u^{r}\right).$

(14)

Let ${\rm Tr}(A)$ denote the trace of a square matrix $A$. Then by definition of ${\rm Tr}$, the following result was shown in [8].

An interesting point is that $\Phi_{r}^{(\infty)}({\bf 0})$ is the return “matrix weight” at time $r$ for the walk on not $T_{N}^{d}$ but $\mathbb{Z}^{d}$. We should remark that in general ${\rm Tr}(\Phi_{r}^{(\infty)}({\bf 0}))$ is not the same as the return probability at time $r$ for QW and CRW, but for RW.

Furthermore, we introduce

$\displaystyle C_{r}(A,T^{d}_{\infty})=\lim_{N\to\infty}C_{r}(A,T^{d}_{N}).$

Therefore, by using the above equation, Theorem 2, and Eq. (14), we have

From now on, we will present the result on only ${\cal L}\left(A,T_{\infty}^{d},u\right)$ and $C_{r}(A,T^{d}_{\infty})$, since the corresponding expression for “without $\lim_{N\to\infty}$” is the essentially same (see Theorems 1 and 3, for example).

Next, we consider a relation between the logarithmic zeta function ${\cal L}\left(A,T_{\infty}^{d},u\right)$ for two-state QWs on the one-dimensional torus $T_{N}^{1}$.

First we deal with general walks including QWs on the one-dimensional torus $T^{1}_{N}$ whose $2\times 2$ coin matrix $A^{(m)}$ (M-type) or $A^{(f)}$ (F-type) as follows:

$\displaystyle A^{(m)}=\begin{bmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix},\qquad A^{(f)}=\begin{bmatrix}a_{21}&a_{22}\\ a_{11}&a_{12}\end{bmatrix},$

since

$\displaystyle A^{(f)}=\left(I_{1}\otimes\sigma\right)A^{(m)}=\sigma A^{(m)}=\begin{bmatrix}0&1\\ 1&0\end{bmatrix}\begin{bmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix}.$

Set $k=k_{1}$ and $\widetilde{k}=\widetilde{k}_{1}$. In this case, we take

$\displaystyle P_{1}=\begin{bmatrix}1&0\\ 0&0\end{bmatrix},\qquad P_{2}=\begin{bmatrix}0&0\\ 0&1\end{bmatrix}.$

Thus we immediately get

	$\displaystyle\widehat{M}_{A^{(m)}}(\widetilde{k})$	$\displaystyle=e^{i\widetilde{k}}P_{1}A^{(m)}+e^{-i\widetilde{k}}P_{2}A^{(m)}=\begin{bmatrix}e^{i\widetilde{k}}a_{11}&e^{i\widetilde{k}}a_{12}\\ e^{-i\widetilde{k}}a_{21}&e^{-i\widetilde{k}}a_{22}\end{bmatrix},$		(15)
	$\displaystyle\widehat{M}_{A^{(f)}}(\widetilde{k})$	$\displaystyle=e^{i\widetilde{k}}P_{1}A^{(f)}+e^{-i\widetilde{k}}P_{2}A^{(f)}=\begin{bmatrix}e^{i\widetilde{k}}a_{21}&e^{i\widetilde{k}}a_{22}\\ e^{-i\widetilde{k}}a_{11}&e^{-i\widetilde{k}}a_{12}\end{bmatrix}.$		(16)

By these equations, we have

$\displaystyle\det\Big{(}I_{2}-u\widehat{M}_{A^{(s)}}(\widetilde{k})\Big{)}=1-{\rm Tr}\left(\widehat{M}_{A^{(s)}}(\widetilde{k})\right)u+\det\left(\widehat{M}_{A^{(s)}}(\widetilde{k})\right)u^{2}\qquad(s\in\{m,f\}).$

Then the result given in [8] can be rewritten in terms of the logarithmic zeta function ${\cal L}\left(A,T_{\infty}^{d},u\right)$ as follows:

Remark that Proposition 1 is also obtained by Theorem 1.

From now on, we focus on QWs in one dimension. One of the typical classes of QWs for $2\times 2$ coin matrix $A^{(m)}$ (M-type) or $A^{(f)}$ (F-type) is as follows:

$\displaystyle A^{(m)}=\begin{bmatrix}\cos\xi&\sin\xi\\ \sin\xi&-\cos\xi\end{bmatrix},\qquad A^{(f)}=\begin{bmatrix}\sin\xi&-\cos\xi\\ \cos\xi&\sin\xi\end{bmatrix}\qquad(\xi\in[0,2\pi)).$

When $\xi=\pi/4$, the QW becomes the so-called Hadamard walk which is one of the most well-investigated model in the study of QWs. Then the result given in [8] can also be rewritten in terms of the logarithmic zeta function like Proposition as follows:

Note that the result on ${\cal L}\left(A^{(s)},T_{\infty}^{1},u\right)$ in Proposition 2 is also derived from Proposition 1. Here the following result holds.

Now, we consider a relation between the logarithmic zeta function ${\cal L}\left(A,T_{\infty}^{d},u\right)$ for RWs on the higher-dimensional torus $T_{N}^{1}$, see [13].

From definition of the (simple symmetric) RW on $T^{d}_{N}$ (see [23, 27]), we easily see that

	$\displaystyle{\rm Spec}\left(P^{(D,c)}\right)$	$\displaystyle=\left\{\frac{1}{d}\sum^{d}_{j=1}\cos\left(\frac{2\pi k_{j}}{N}\right)\bigg{|}\ k_{1},\ldots,k_{d}\in\mathbb{K}_{N}\right\}$
		$\displaystyle=\left\{\frac{1}{d}e^{(d,\cos)}(\widetilde{\bm{k}})\bigg{|}\ k_{1},\ldots,k_{d}\in\mathbb{K}_{N}\right\},$		(17)

where $P^{(D,c)}$ is the transition probability matrix of the (simple symmetric) RW on $T^{d}_{N}$. Here the RW on $T^{d}_{N}$ jumps to each of its nearest neighbors with equal probability $1/(2d)$. Noting Eq. (17), the result given in [13] can be rewritten in terms of the logarithmic zeta function as follows: