Restricted random walks

This corresponds to a random walk in the presence of a boundary. Here we list some examples:

• Traps: Consider a 1d lattice which has ``traps'' at sites $x=0$ and $x=l$. A random walker starts a site $x_0$ somewhere in the center ($0 < x_0 < l$) and steps to a neighboring site with equal probability. It can be shown that the mean ``fist passage time'' $\tau$ for a particle to be trapped is given by
  
  $$\tau=\frac{1}{2D}x_0(l-x_0)$$
  
  
  where $D$ is the self-diffusion constant in the absence of the traps, and the average is over all possible walks.

• Reflecting boundaries: Consider the case of reflecting sites at $x=0$ and $x=l$. If a walker reaches a boundary, it is reflected back to the previous site. What is the probability of finding a walker in site $x$ after $N$ steps?

• ``Gambler's ruin'': Suppose that two gamblers are given $10 and on each throw of a dice, one gambler must win $1 and the other must loose $1. How long can they play in average until the capital of the loser is exhausted?