  As an extension of Polya's classical result on random walks on the square
grids ($\Z^d$), we consider a random walk where the steps, while still have
unit length, point to different directions.
  We show that in dimensions at least 4, the returning probability after $n$
steps is at most $n^{-d/2 - d/(d-2) +o(1)}$, which is sharp. The real surprise
is in dimensions 2 and 3.
  In dimension 2, where the traditional grid walk is recurrent, our upper bound
is $n^{-\omega (1)}$, which is much worse than higher dimensions.
  In dimension 3, we prove an upper bound of order $n^{-4 +o(1)}$. We discover
a new conjecture concerning incidences between spheres and points in $\R^3$,
which, if holds, would improve the bound to $n^{-9/2 +o(1)}$, which is
consistent % with the $d \ge 4$ case. to the $d \ge 4$ case. This conjecture
resembles Szemer\'edi-Trotter type results and is of independent interest.
