suggests that we can define a continuous time in the $N_{\rm X} \to \infty$ limit.

![The eigenvalues $\alpha_{p} \ (p=1,\ldots,N_{\rm X})$ of $X_{0}$ are plotted against the label $p$ in the ascending order for a typical solution with $N_{\rm X}=64$. []{data-label="fig:X0"}](X0.eps){width="10cm"}

![The quantity $\Delta_{pq}$ defined by Eq. (\[eq:ampX\]) is plotted against the labels $p$ and $q$ for the typical solution with $N_{\rm X}=64$. []{data-label="fig:ampX"}](M_2.eps){width="10cm"}

In the basis that makes $X_{0}$ diagonal, the spatial matrices $X_{i} \  (i=1,2,3)$ are not diagonal in general, but they actually turn out to be band-diagonal. In order to see it, we plot $$\label{eq:ampX}
\Delta_{pq} 
=\sum_{i=1}^{3}\left|(X_{i})_{pq}\right|^{2}$$ for the same solution in Fig. \[fig:ampX\]. We find that $\Delta_{pq}$ becomes very small for $|p-q|\geq n$ with some integer $n$, which is $n \sim 10$ for the configuration in Fig. \[fig:ampX\]. The band-diagonal structure guarantees the locality of time, which enables us to extract the time evolution as we discuss in the next section.

The uniform distribution of $\alpha_p$ and the band-diagonal structure in $X_i$ are common features of the solutions we obtain. They are actually shared also by the dominant configurations generated by previous Monte Carlo studies[@Ito:2013ywa; @Ito:2015mxa; @Ito:2017rcr; @Azuma:2017dcb]. As we mentioned below (\[eq: EOM\]), the classical 