# The plane is too small 2

Let $\{X_n|n\in\mathbb{N}_0\}\subset\mathbb{Z}^2$ be the symmetric random walk in $2$$d$, 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.
Consider a permutation $\sigma$ of $[n]$, and let $X^n_j$ be the random variable that takes the value $1$ if this permutation fixes $j$, and takes the value zero elsewhen. Set $X_n= \sum X_n^j$.
1. Express $P(X_n=k)$ in terms of $P(X_m=0)$ for a suitable $m$.
2. Find $P(X_n >0)$, and hence find $P(X_n=0)$.
3. Find $P(X_n=0)$ as $n\to\infty$, and deduce that $X_n$ converges in distribution to a Poission variable. Of what weight?