Let $G$ be the Green function of the simple random walk on $\mathbb{Z}^d,\:d\geq 3$; i.e. $$G(x) = E \sum_{i=0}^{+\infty} 1_{X_i=x},$$ where $X$ is the simple random walk starting from $0$. The behaviour of $G(x)$ is well know when $x$ is large. But do we know how the function behaves when $x$ is small?

I tried to look in book of Spitzer (Principles of RW) and Lawler, Limic (Random Walk: Modern Introduction). I only found "inverse Fourier transform" formula which I do not have idea how to use (besides some numeric calculations).

May be some of you know other references?

My question stems from esimation of $$S_d:=\sum_{x\in \mathbb{Z}^d\setminus \lbrace 0 \rbrace} a_x \left(\sqrt{1+\frac{G(x)^2}{G(0)}}-1\right),$$ where $a_x$ are some coefficients. In the simplest case $a_x =1$ (or $a_x$ = 1 if $\sum_{i=1}^d x_i$ is odd and -1 otherwise; "chessboard of -1, 1").

In the case of $a_x=1$ one can easily prove that $S=\infty$ if $d=3,4$ and $S<\infty$ if $d\geq5$ (just using the fact that $G(x) \sim |x|^{2-d}$ when $x$ is large.)

But is it possible to say anything more about $S_d$ when $d>5$? E.g. I expect that $S_d \rightarrow 0$ as $d\rightarrow +\infty$.

**I made a mistake in the first version the sum was over $\mathbb{Z}^d$ and it should be over $\mathbb{Z}^d\setminus \lbrace0\rbrace$.**

**Update 24.05.2011** Thanks to observations made in Didier Piau proof I was able to prove that indeed $S_d \rightarrow 0$ (in the case when $a_x=1$). I put the proof here.