A friend of mine recently asked me if there was a reason why, when he put a bed sheet in his washing machine, it seemed like the an abnormal proportion of the content of the machine ended up wrapped inside the bed sheet (even if the bed sheet was separately put inside it).
Now I'm not actually sure this is a accurate statement (and I don't actually care so much), but my first instinct was to try to think of it as a statement on percolation clusters. However these clusters are very much non-homeomorphic to the plane. Hence it made me wonder:
Question: is there any literature on "random sheets", i.e. a random map from the plane to space which is (say) Lipschitz?
Ideally the image should be non-self-intersecting, but that sounds already too complicated. As a side note, looking at a random field on (a lattice of) the plane is not fitting here, since there are conditions which are induces by loops, and not just by distance. Furthermore, if one maps the whole plane, one would expect the image to be unbounded (and this is not quite the case in random fields).