# 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.

• Although I am no expert of this subject, this model would be extremely hard to analyze mathematically due to its non-Markovian nature. Here are some of my random thoughts: (1) for $k = 1$, similar models have been studied, such as self-avoiding walk (SAW), non-backtracking walk (NBW), etc. Physicists seem to have certain predictions on the case $k=1$, although I rarely know any. (2) The local time field of a random walk will look like gaussian free field, so if $k$ is very large, the work will behave as if it avoids level sets of GFF, which is known to be very fractal-like. – Sangchul Lee Feb 9 at 22:49
• Thanks for your comment! I'm aware that the failure of the Markov-property is a key difficulty and I also know that $k=1$ is a SARW. However, I am not up-to-date about what is known even in that case. (2) sounds interesting. – Mars Plastic Feb 9 at 23:09
• arxiv.org/abs/math/0611734 this paper on “RW with collapsing bonds” seems relevant. But the “ice” is repaired in a queue setting and rather being broken after stepping on it $k$ times, it breaks with probability $p>0$. The authors obtain a scaling limit related to Brownian motion. Indeed there is non-Markovian aspects but this setting (with repairable bonds) is still amenable. – LoveTooNap29 Mar 12 at 18:06
• @LoveTooNap29 This is really neat. Basically, they assume that it is cold enough for the holes in the ice to freeze over again after some time. – Mars Plastic Mar 12 at 18:19
• @MarsPlastic Glad to share! Indeed that is a nice way to reinterpret in terms of the ice idea. – LoveTooNap29 Mar 12 at 20:11

$$(1)$$ 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.