Random walk returning probability
this ref covers it. 


  In sections 5 &amp; 6 we use the elliptic integral to express the probability, less than one, that certain biased random walks return to the origin. The probability of a return to the origin for these walks can then be computed accurately and easily using Gauss' arithmetic-geometric mean (AGM) method to evaluate the elliptic integral.


[1] Recurrence of Simple Random Walk in the Plane
Terence R. Shore and Douglas B. Tyler
The American Mathematical Monthly
Vol. 100, No. 2 (Feb., 1993), pp. 144-149