Malo Hillairet

Throw a bunch of rectangular tiles of dimensions 1x1 and 1x2 (which may be rotated by a quarter turn) inside a planar frame, and use these to pave the region delimited by the frame. What do we expect the paving to look like, do the 1x2 tiles tend to align with each other? What if we throw a lot more 1x2 tiles than 1x1 tiles? A way to approach these questions is to consider infinite-volume measures, that is, random pavings of the entire space. We will see that it is closely related to the so-called "phase transition" phenomena, which naturally appear in physics, such as solid/liquid/gas transitions for the states of matter.