A particle starts at an origin $O$ in three-space. Thinking of point $O$ as the center of a cube 2 units on a side. One move in this walk sends the particle with equal likelihood to one of the eight corners of the cube. That is to say, at every walk, the particle has a 50/50 chance of moving one unit up or down, one unit left or right, and one unit front or back. If this walk continues infinitely, what is the probability that the particle returns to $O$?
So far, I thought of the following:
$$P(\text{particle is at $O$ after $2n$ walks})=\left(\binom{2n}{n}\left(\dfrac{1}{2}\right)^{2n}\right)^3$$
I then applied Stirling's approximation: $n!\sim\sqrt{2\pi}n^{n+\frac{1}{2}}e^{-n}$ to get
$$P(\text{particle is at $O$ after $2n$ walks})\approx\left(\dfrac{1}{\pi n}\right)^{\frac{3}{2}}$$
I then tried to sum this:
$$P(\text{particle returns to $O$})=\sum_{n=1}^\infty \left(\dfrac{1}{\pi n}\right)^{\frac{3}{2}}\approx 0.47 $$
Unfortunately, this answer is incorrect and I suspect that there is a problem with the values of my summation.
Best Answer
The number of ways to get back to $O$ after $2n$ moves is given by
$$\begin{align} \sum_{r+s+t=n}\binom{2n}{r,r,s,s,t,t} &=\sum_{r+s+t=n}\binom {\color{magenta}{2n}}{\underbrace{r,s,t}_{\color{magenta}n},\underbrace{r,s,t}_{\color{magenta}n}}\\ &=\sum_{r+s+t=n}\left[\color{magenta}{\binom {2n}n}\binom nr\binom {n-r}s\color{lightgrey}{\binom {n-r-s}t}\right] \left[\color{magenta}{\binom nn}\binom nr\binom {n-r}s\color{lightgrey}{\binom {n-r-s}t}\right]\\ &=\binom {2n}n\sum_{r=0}^n\binom nr ^2 \;\sum_{s=0}^{n-r}\binom {n-r}s ^2\\ &=\binom {2n}n\sum_{r=0}^n \binom nr ^2\binom {2(n-r)}{n-r}\\ &=\color{red}{\binom {2n}n\sum_{r=0}^n \binom nr ^2\binom {2r}r\qquad\blacksquare}\end{align}$$
Hence, the probability of getting back to $O$ after $2n$ moves is given by $$\color{red}{\frac 1{6^{2n}}\binom {2n}n\sum_{r=0}^n\binom nr^2\binom {2r}{r}}$$
NB: There does not seem to be a closed form for $(*)$. See this OEIS series here. See also this note here on random walks.
(Previous answer below) $$\tiny\begin{align} \sum_{r+s+t=n}\binom {2n}{r,s,t,r,s,t} &= \sum_{r+s+t=n}\binom {2n}r \binom {2n-r}s\binom {2n-r-s}t\binom nr\binom {n-r}s\overbrace{\binom {n-r-s}t}^1\\ &=\sum_{r+s+t=n}\binom{2n}{2n-r}\binom {2n-r}{\color{orange}{2n-r-s}}\binom {\color{orange}{2n-r-s}}n\binom nr\binom {n-r}s\\ &=\sum_{r+s+t=n}\binom {2n}{\color{blue}{2n-r}}\binom {\color{blue}{2n-r}}{\color{}n}\binom {n-r}{n-r-s}\binom {n}r\binom {n-r}s\\ &=\binom {2n}n\sum_{r+s+t=n}\binom n{n-r}\binom {n-r}{s}\binom nr\binom {n-r}s\\ &=\binom {2n}n\sum_{r+s+t=n}\binom nr^2\binom {n-r}s^2\\ &=\binom {2n}n\sum_{r=0}^n\binom nr^2\sum_{s=0}^{n-r}\binom{n-r}s^2\\ &=\binom {2n}n\sum_{r=0}^n\binom nr^2\binom {2(n-r)}{n-r}\\ &=\binom {2n}n\sum_{r=0}^n\binom nr^2\binom {2r}{r}\tag{*}\\ \end{align}$$