How many times will a random walk cross a boundary line if permitted to continue walking forever? A simple random walk on formula_1 will cross every point an infinite number of times. This result has many names: the "level-crossing phenomenon", "recurrence" or the "gambler's ruin". The reason for the last name is as follows: a gambler with a finite amount of money will eventually lose when playing "a fair game" against a bank with an infinite amount of money. The gambler's money will perform a random walk, and it will reach zero at some point, and the game will be over.