Reconstructing random jigsaws

TL;DR

This paper investigates the reconstructibility of random edge-coloured grids, establishing thresholds for the number of colours needed for high probability of unique reconstruction as grid size grows.

Contribution

It improves previous bounds by proving sharp thresholds for reconstructibility in random grid colourings, showing it occurs with high probability when colours are proportional to grid size.

Findings

Reconstructibility probability tends to 1 if q ≥ Cn.

Reconstructibility probability tends to 0 if q ≤ cn.

Sharp phase transition thresholds are established for random grid colourings.

Abstract

A colouring of the edges of an $n \times n$ grid is said to be \emph{reconstructible} if the colouring is uniquely determined by the multiset of its $n^2$ \emph{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 $n \times n$ 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 $q = o(n^{2/3})$ and from above, Bordenave, Feige and Mossel and Nenadov, Pfister and Steger independently showed, for any fixed $\epsilon > 0$, that such a colouring is reconstructible when $q \ge n^{1+\epsilon}$. Here, we improve on these results and prove the following: there exist absolute constants $C, c > 0$ such that, as $n \to \infty$, the probability that a random colouring as above is reconstructible tends to $1$ if $q \ge Cn$ and to $0$ if $q \le cn$.