Random walk returning probability Consider a two-dimensional random walk, but this time the probabilities are not $1/4$, but some values $p_1, p_2, p_3, p_4$ with $\sum_{i=1}^4 p_i=1$. For example, from $(0,0)$, it goes to $(1,0)$ with $p_1$, goes to $(0,1)$ with $p_2$ etc. 

I am interested in the probability $x$ of going back to (0,0), starting from (0,0). In general, this probability is not 1 (I guess that $x$ is not rational in general). The question is, given a probability threshold $r$, is it decidable that $x\geq r$?
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