I'm considering a random walk $x_t$ on an array of $N$ sites. At each time $t$, the process jumps to site $y$ with a probability proportional to a given distribution $f\left(y- x_{t-1}\right)$.
The complication is that this random walk cannot visit the same site twice, and cannot exit the array. As the array gets filled up, therefore, there will be fewer and fewer jumps that it can actually make.
The function $f(z)$ is the distribution of the jumps without the constraint. Let's call $g_t(z)$ the actual distribution of the jumps, that, is the distribution of the variable $z_t=x_t-x_{t-1}$.
$f$ is independent on time; but because of the constraints, the actual distribution $g_t$ of the jumps will be time dependent.
I'm trying to understand how $g_t$ will be skewed with respect to $f$. If anybody has any suggestion, I would very much appreciate it.