The role of eigenvectors in understanding absorbing Markov chains

Suppose a friend of yours is gambling one dollar at a time in a casino (and he gives one dollar to Sanchit every time he gets to 1000000 dollars, so he can get a beer and rest before starting to play again). If you join him after a "long" time, and he isn't broke yet, how much money do you expect him to have? And what do we mean by "long"? And what is Sanchit doing in a casino? We will get some bounds for some of these questions in terms of how a certain eigenvector oscillates.