Random walk on thin ice? My Question: Is the stochastic process which is described below a (special case of a) well-studied model? What kind of properties are known under which assumptions?
Let's think of a random walker on a frozen lake. Obviously, this lake is a subset $E$ of $\mathbb Z^n$. The ice isn't too stable, though, and therefore our poor walker can occupy each vertex only $k\in\mathbb N$ times before the ice collapses. Luckily, the walker is quick and agile enough to leave the vertex before she is swallowed by the lake. However, she will no longer be able to visit this vertex, changing the rules for the future of her odyssey.
I know that there is a huge community of probabilists who investigate random walks in random environments (and also environments that are randomly changing over time), so I'd be rather surprised if there were no results about this kind of process. The above description is intenionally vague: One might think of different asymmetry properties and different ways in which the "weight" of a removed vertex will be given to neighbouring (or maybe even more) vertices. The "stability" $k$ might depend on the site, and it might also be random. Lots of directions in which one might take this...
Obvious questions about this process: What's the probability that the walker will eventually get trapped between collapsed vertices? If e.g. the lake $E$ is finite, the probability is obviously one. What about the expected time for this to happen? If $E=\mathbb Z^n$, which recurrence properties (or general asymptotic properties) does the process have and in which sense?
I would appreciate it very much, if anyone could give me a hint where to look for answers.
PS: For the inspiration for this question, look here.
 A: Possibly more of an extended comment than an answer, but a classic paper is Lehman and Weiss [1] In this paper it is proved that, for a restricted (where "restricted" refers to the thin ice problem in your language) walk on the 2D lattice,
$(1)$ The probability the random walker survives $n$ steps without being trapped decreases exponentially in $n$. 
$(2)$ The random walker is trapped at some point almost surely (which follows from $(1)$.)
$(3)$ The expected number of steps before getting trapped is finite (also follows from $(1)$.)
They remark that $(1)$ holds for higher dimensional lattices with arbitrary probabilities. The argument is essentially that the walker always has a non-zero probability of being trapped on its first visit to a given hypercube in the lattice. In 2D, this looks like a 'G' shape. I don't see why this wouldn't hold if $k > 1$. Calculating the expected number of steps before getting trapped is completely beyond me though.
[1] Lehman, R. Sherman, and George H. Weiss. “A Study of the Restricted Random Walk.” Journal of the Society for Industrial and Applied Mathematics, vol. 6, no. 3, 1958, pp. 257–278. JSTOR, www.jstor.org/stable/2098696. 
