For all $k \in \llbracket 1,n \rrbracket$, we denote by $C_{n,k}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by $$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad C_{n,k}(i,j) = \begin{cases} 1 & \text{if } (i \in \llbracket 1,k \rrbracket \text{ and } j = i+n-k) \text{ or } (i \in \llbracket k+1,n \rrbracket \text{ and } j = i-k) \\ 0 & \text{otherwise} \end{cases}$$ We note that $C_{n,n} = I_n$. We set $J_n^{(1)} = C_{n,1} + C_{n,n-1}$. Verify that, in the case where $n = 9$, $J_n^{(1)}$ is the matrix such that, for all $(i,j) \in \llbracket 1,9 \rrbracket^2$, the coefficient with index $(i,j)$ equals 1 if the vertices $i$ and $j$ of the graph are connected by an edge and equals 0 otherwise. This means that each particle interacts only with its two nearest neighbors.
For all $k \in \llbracket 1,n \rrbracket$, we denote by $C_{n,k}$ the matrix of $\mathcal{M}_n(\mathbb{R})$ defined by
$$\forall (i,j) \in \llbracket 1,n \rrbracket^2, \quad C_{n,k}(i,j) = \begin{cases} 1 & \text{if } (i \in \llbracket 1,k \rrbracket \text{ and } j = i+n-k) \text{ or } (i \in \llbracket k+1,n \rrbracket \text{ and } j = i-k) \\ 0 & \text{otherwise} \end{cases}$$
We note that $C_{n,n} = I_n$. We set $J_n^{(1)} = C_{n,1} + C_{n,n-1}$.
Verify that, in the case where $n = 9$, $J_n^{(1)}$ is the matrix such that, for all $(i,j) \in \llbracket 1,9 \rrbracket^2$, the coefficient with index $(i,j)$ equals 1 if the vertices $i$ and $j$ of the graph are connected by an edge and equals 0 otherwise. This means that each particle interacts only with its two nearest neighbors.