Imagine a discrete random walk on an infinite one-dimensional lattice where, for every unit interval of time, $(t_1, t_2, ...)$, the walker takes a step with uniform probability to its left or right.

We add the constraint that each time the walker visits a vertex, the vertex is transiently "blocked", and cannot be revisited by the walker until it is "unblocked", which has a constant probability $p$ of occurring during every unit interval of time $t_i$ prior to the walker taking a step. In other words, every time the walker visits a vertex, the vertex is assigned an exponentially distributed random variable $X$ (always with the same rate parameter $\lambda = p$) that determines the number of time intervals that need to pass (counting from the interval in which the site is "blocked") before it can be reoccupied by the walker. If the vertex to the right or left of a walker is blocked during some time interval, it will move to the left or right, respectively. In the situation where sites to the left and right of the random walker are "blocked", the walker will remain in place for that unit interval of time.

To provide the extremal examples, if $p = 1$, sites are immediately unblocked prior to the walker taking its next step, and one will have a vanilla one-dimensional random walk with a Gaussian probability distribution. If, however, we set $p = 0$, sites will never "unblock" and the walk will become fully self-avoiding and, with a direction chosen with uniform probability, continuously move to the right or left without ever revisiting the origin.

As a function of $p$, what is the probability distribution for this walker? How does this walk generalize to higher dimensions, specifically $D = 2$? Is there a "magic term" for this sort of walk, and are there any good literature references?

As a sort of cute extension, provided some $p$ can we predict the mean distance the walker will travel returning to its point of origin?

Note - I would be happy to accept the simplification where, instead of assigning vertices an exponentially distributed random variable, $X$ can adopt an alternative probability distribution.