If the steps are iid uniform as in the title, is the return probability known? Is it positive? Answers, comments, references welcome. Clearly each of these steps is not equivalent to $d$ steps of type $\pm e\_i$, especially for large $d$.


Its distribution after $n$ steps is that of $d$ independent uniform $\pm 1$ sums isn't it?



Does walk on $Z^d$ with steps $(\pm 1,\pm1,\ldots,\pm 1)$ return to origin?
These random walks are recurrent when $d\le 2$ and transient when $d \ge 3$. That behavior happens for a wide variety of random walks.


The expected number of returns to the origin is $$\sum\_{n=1}^\infty \left(\frac{1}{2^{2n}}{2n \choose n}\right)^d.$$


If the expected number of returns is $x \lt \infty$, then the probability of returning is $1-\frac{1}{x+1}$. 


Mathematica evaluates the probability of returning for $d=3$ as 


$$1-\frac{\Gamma(3/4)^4}{\pi}$$


but for greater $d$ the methods I tried in Mathematica left it as a particular value of a hypergeometric function which restates the series.


