Proof about Simple random walk in $\mathbb{R}^{d}$ I have read something about random walks in $\mathbb{R}^{d}$.
The random walks is assumed to be started at origin.
There is a theorem said that
For $d =1$ or $2$, the random walk is recurrent.
(i.e. almost all path return origin infinite many time) 
For $d\geq3$, almost all path returns the origin at most finite number of time. And there is a positive probability that the path never returns to the origin.
I don't want to give a rigorous definition of random walk since my question is not really related to the definition.
In the proof, we have $\hat{\mu}(\xi) = \frac{1}{d}\sum\limits_{k=1}^d cos(2\pi\xi_k)$ for $\xi\in\mathbb{R}^{d}$ and $Q=\{\xi\in\mathbb{R}^{d}|-\frac{1}{2}<\xi_{j}\leq\frac{1}{2},j=1,...,d\}$
The proof said that:
Now since$$1-\hat{\mu}(\xi)= 1-\frac{1}{d}\sum\limits_{k=1}^d cos(2\pi\xi_k) = \frac{2\pi^2}{d} |\xi|^2 +O(|\xi|^4)\:\:as\:|\xi|\to 0$$
and $1-\hat{\mu}(\xi)\geq c_1$ if $|\xi|\geq c_2$ and $\xi\in Q$, for some suitable positive constants $c_1$ and $c_2$, we have the integral$$\int_{Q} \frac{d\xi}{1-\hat{\mu}(\xi)}$$ diverge for $d=1$ or $2$ but coverge for $d\geq3$
My question is how to find such constants $c_1$ and $c_2$ and how to make such conclusion.
 A: The meaning of your proof is mentioned in the comment, so I just complete it with some details.
Firstly, note that $\hat{\mu(\xi)}$ is just a sum of cosine functions. By the continuity of the $\hat{\mu(\xi)}$ on $Q$ and $\hat{\mu(\xi)} < 1$ for $x \neq 0$, there exists $c_1, c_2 > 0$ such that if $\left|\xi\right| > c_2$, then $\sum_{n = 1}^d \cos \xi_n < d(1 - c_1)$.
Secondly, the convergence of the integral $\int_Q \frac{d\xi}{1 - \hat{\mu(\xi)}}$ can be deduce from the convergence of $\int_B \frac{dx}{\left|x\right|^2}$ for some small ball $B = B_{c_2}(0)$ because $1 - \hat{\mu(\xi)} \geq c_1$ for $\left|\xi\right| \geq c_2$. Now, to evaluate the later integral, by using the polar coordinates, we see that $$\int_{B_1} \frac{dx}{\left|x\right|^2} = C(d)\int_{-1}^1\left|r\right|^{d - 3}dr$$ for some constant $C(d)$ depending on the dimension $d$ only (the constant $C(d)$ and the extra power $d - 1$ of $r$ comes from the use of polar coordinates). Clearly the right integral converges for $d \geq 3$ and diverges for $d = 1, 2$ and this completes your proof.
