The plane is too small 2

Let \{X_n|n\in\mathbb{N}_0\}\subset\mathbb{Z}^2 be the symmetric random walk in 2d, starting at the origin. So, X_0=(0,0) and the transition probabilities are p_{i,j}=1/4 if |i-j|=1 and 0 otherwise.

Let X_n^{+},X_n^{-} be the orthogonal projections of the random walk onto the diagonals \{y=x\} and \{y=-x\}.

  1. Prove that X_n^{+},X_n^{-} are independent symmetric random walks in \mathbb{Z}/2 such that X_n=0 if and only if X_n^{+}=0=X_n^{-}.
  2. Prove that p_{00}^{(2n)} = (\binom{2n}{n}2^{-2n})^2 \sim \frac{A}{n}, as n\to\infty, where A \in \mathbb{R}_{>0} and p_{00}^{(2n)}=P(X_{2n}=(0,0)|X_0=(0,0)). Conclude that \sum_{n=0}^{\infty} p_{00}^{(n)} = \infty. This is equivalent to the recurrence of the random walk.
The plane is too small 2