I've read that Polya coined the term "Random Walk." He analyzed the 1-dimension example and proved that the chances of returning to any point on the line is ultimately 100%. This is how one can think of a gambler's ruin: a person playing a fair random game against a casino will eventually lose all his money.
Now, Polya also showed that this scheme breaks down in higher dimensions. For instance, in the 3 dimensional lattice, the player has a lower chance of going back to its starting point (0.34, although I realized in my research that there is no closed form answer for higher dimensions).
Question: how can we understand the gambler analogy in the context of Polya's higher dimension examples? Does the case for the 3-dimensional lattice mean that for example somehow playing in 3 casinos with 3 different currencies lowers the risk of ruin?
p.s. This is not for gambling. I have never played at a casino, nor intend to... I'm just fascinated with the topic and have researched it as much as I could as a non-mathematician. Thanks for any insights!
In one dimension, "returning to the origin" and "running out of money" are one and the same. In more dimensions, "returning to the origin" corresponds to "simultaneously having a balance of zero dollars at every casino". In contrast, ruin might be thought of as having a net balance of zero dollars. Ruin still happens with 100% probability.