Execute a random walk from the origin on the integer lattice, but bias
the four compass-direction probabilities from $\frac{1}{4}$ each to prefer
to step in a spiraling direction.
Calling the four step vectors $c_0,c_1,c_2,c_3$, with 
$c_i=(\cos (i \, \pi/2), \sin (i \, \pi/2))$, adjust the probabilities as follows.
Let $v$ be the vector from the origin to the last point on the path,
and $n$ the unit normal to $v$, counterclockwise $90^\circ$ to $v$.
Then select step $c_i$ with probability $\frac{1}{4} (1+ c_i \cdot n)$.
<hr />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; [![SpiralVecs][1]][1]
<br />
&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; 
<sup>
$\theta=60^\circ$. $\frac{1}{4}(1+c_2 \cdot n) = (1+\sqrt{3}/2)/4 \approx 0.47$.
</sup>
<hr />
Unsurprisingly, the random walks spiral around the origin:
<hr />
&nbsp; &nbsp; [![3RandWalks][3]][3]
<br />
&nbsp; &nbsp; 
<br />
&nbsp; &nbsp;  [![5RandWalks][2]][2]
<br />
<sup>
$2000$-step random walks. Origin: green. Last point: red.
Three examples, followed by five examples at reduced scale.
</sup>
<hr />

> ***Q***. Does Pólya's recurrence theorem hold for these walks? Do the walks
return to the origin with probability $1$?

All but one of the above examples (the penultimate) returned to the origin, but usually rather quickly.


  [1]: https://i.sstatic.net/RiYKa.jpg
  [2]: https://i.sstatic.net/NNh33.jpg
  [3]: https://i.sstatic.net/W3pUv.jpg