I have a question:

Suppose a knight makes a "casual walk" on an infinite chessboard. In particular, the knight follows the standard chess rules each round and moves with a probability of 1/8 to one of his eight available boxes.

What is the probability that the jumper will be back on the starting field? How many trains are to be expected?

Here is a simulation: We can see that knights always return with an even number, but not with an odd number. http://varianceexplained.org/r/knight-chess/

It's just a simple, random walk on an infinite graph. So it is also recurring after a series of moves. Denote the $ X_n $ be the position of the knight in steps $ n $, It is asked

$$ mathbb {P} _0 ( tau_n < infty) =? text {and} mathbb {E} _0 ( tau_n < infty) =? $$

Where $ tau_n: = {n geq 1: X_n = 0 } $ and $ 0 $ is the starting point.

Is there a precise solution to get the result?