Forgive this simple-minded approach, but it may be instructive when made rigorous by someone who knows what they are doing.

The probability of going east-west is 1/2, as is going north-south. However, most of the time, the probability for increasing the distance from the origin is greater than 1/2. This is because one of (say) east west is biased toward increasing distance, while the other (say) is half increasing and half decreasing. So for near the origin, there is a chance of returning, but (as Martin Hairer suggests) once you get far enough away, the dynamic seems transient.

I turn the calculation of "far enough away" over to those with more experience. I suspect the probability of returning to within S distance of the origin given one is at distance R in this dynamic is exponential in S-R (I assume S less than R).

Gerhard "Wonders About Inward Spiral Dynamics" Paseman, 2017.06.23.

Forgive this simple-minded approach, but it may be instructive when made rigorous by someone who knows what they are doing.

I think Anthony Quas's suggestion of looking at radius good, but I do not understand his use of the word 'recurrent'. The probability of going east-west is 1/2, as is going north-south. However, most of the time, the probability for increasing the distance from the origin is greater than 1/2. This is because one of (say) east west is biased toward increasing distance, while the other (say) is half increasing and half decreasing. So for near the origin, there is a chance of returning, but (as Martin Hairer suggests) once you get far enough away, the dynamic seems transient.

I turn the calculation of "far enough away" over to those with more experience. I suspect the probability of returning to within S distance of the origin given one is at distance R in this dynamic is exponential in S-R (I assume S less than R).

Gerhard "Wonders About Inward Spiral Dynamics" Paseman, 2017.06.23.