Given a function $f : \mathbb Z\times \mathbb Z \rightarrow \mathbb{R}$ we define the discrete laplacian of $f$, $\triangle_df$, by the following rule
$\triangle_df(x,y)= \dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, y - 1)}{4}-f(x,y)$
In order to prove recurrence of the simple random walk in $\mathbb{Z}^2$ I need to construct a superharmonic function for big values of $|(x,y)|$. To do this, Varadhan at example 5.7, suggest one to prove that $\triangle_d \log |(x,y)| \leq \dfrac{C}{|(x,y)|^4}$ and $\triangle_d \dfrac{1}{|(x,y)|} \geq \dfrac{c}{|(x,y)|^3}$ for big values of $|(x,y)|.$
I've tried to use that
$$\dfrac{f(x + 1, y)+f(x, y + 1) + f(x - 1, y) +f(x, y - 1)}{4}-f(x,y)=$$ $$=a_1\partial_{1}^{2}f(x+b_1,y)+a_2\partial_{2}^{2}f(x,y+b_2)$$
for some constants $a_i,b_i$ and to estimate the last expression. Does someone have any hint?
Thanks in advance