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I am looking for a classical analogue of localization for quantum walks.

First, I draw for each point in $x \in \mathbb{Z}^2$ (with some distribution) the numbers $u_x,d_x,l_x,r_x$ such that $u_x+d_x+l_x+r_x = 1$ independently for each point.

Then I start a random walk such that for every time I come to some point $x$ I go upp, down, left and right with probabilities $u_x, d_x, l_x, r_x$.

Now I ask questions like what the probability is that the walk stays inside some big box, whether it escapes to infinity and so on. This should be a well studied question. Do you have some references?

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    $\begingroup$ A very interesting question indeed... you may also want to consider the generalisation of your setting to Markov chains with random transition weights. In this case, one may ask natural questions about reducibility and/or transience of random stochastic matrices... $\endgroup$
    – Peter Wildemann
    Commented Oct 22, 2020 at 16:37
  • $\begingroup$ For some inspiration: Solomon, Fred. "Random walks in a random environment." The annals of probability (1975): 1-31. $\endgroup$
    – Peter Wildemann
    Commented Oct 22, 2020 at 16:53
  • $\begingroup$ Visit the following page for exploring the "neighbourhood" of above paper. It seems to be quite an active field: connectedpapers.com/main/… $\endgroup$
    – Peter Wildemann
    Commented Oct 22, 2020 at 16:56
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    $\begingroup$ For activity through the early 2000's, you can look up my St Flour lecture notes (Springer) on random walk in random environments. It has a detailed bibliography. For work since then, Sznitman , Bolthausen, Varadhan. In particular, in Z^2 we know (due to Zerner and Merkl) that the probability that a one-dimensional projection escapes to $+\infty$ is either 0 or 1, but there is still no effective criterion for transince. $\endgroup$
    – ofer zeitouni
    Commented Oct 22, 2020 at 20:23
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    $\begingroup$ Incidentally, the paper of Kalikow (1982) is very relevant to the 2D question you are asking, probably more than Solomon's, who deal with 1D. $\endgroup$
    – ofer zeitouni
    Commented Oct 22, 2020 at 20:26

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