Reconstructing Random Jigsaws

A colouring of the edges of an n×n grid is said to be reconstructible if the colouring is uniquely determined by the multiset of its n tiles, where the tile corresponding to a vertex of the grid specifies the colours of the edges incident to that vertex in some fixed order. In 2015, Mossel and Ross asked the following question: if the edges of an n× n grid are coloured independently and uniformly at random using q = q(n) different colours, then is the resulting colouring reconstructible with high probability? From below, Mossel and Ross showed that such a colouring is not reconstructible when q = o(n) and from above, Bordenave, Feige and Mossel and Nenadov, Pfister and Steger independently showed, for any fixed ε > 0, that such a colouring is reconstructible when q ≥ n. Here, we improve on these results and prove the following: there exist absolute constants C, c > 0 such that, as n→∞, the probability that a random colouring as above is reconstructible tends to 1 if q ≥ Cn and to 0 if q ≤ cn.