I’m reading a paper on $2$D discrete Laplace operator, and perhaps because it’s an old paper, the notation in it really bothers me a lot. So can someone please explain it to me? For example, the notation I have highlighted in red line in the screenshot, such as what $G$ and $\log R$ mean and how to interpret these equations?
This paper leaves out so many details that it's understandable that the notation doesn't make sense. The context here is that we want to study functions on the two-dimensional integer lattice $\mathbb Z^2$. In particular, $G$ denotes a real-valued function on $\mathbb Z^2$, where $G_{i,j}$ denotes the value of the function at $(i,j) \in \mathbb Z^2$.
Rather than a single function $G$, this paper studies a family $G(\alpha)$ of functions on $\mathbb Z^2$ parametrized by $\alpha$. So now $G_{i,j}(\alpha)$ denotes the value of the function $G(\alpha)$ at the lattice point $(i, j)$. The function $G_\alpha$ is defined by (1a) to be a "fundamental solution" to a certain Laplacian operator $\mathbf L_\alpha$. In general, a function that satisfies this type of relation to a Laplacian operator is known as a Green's function which explains the choice of notation $G$.
The family of operators $\mathbf L_\alpha$ is a modification of the usual Discrete graph Laplacian $\mathbf L = \mathbf L_0$ on $\mathbb Z^2$ with nearest-neighbor edges. There has been a lot of work on studying the Green's function for this Laplacian, e.g. Malik Mamode, Revisiting the discrete planar Laplacian: exact results for the lattice Green function and continuum limit.