Domino tilings of cylinders: connected components under flips and normal distribution of the twist (Q2227824)

A cubiculated region in \(\mathbb R^3\) is a union of finitely many unit cubes with vertices in \(\mathbb Z^3\). A cylinder is a cubiculated region of the form \(R_N = D \times [0,N]\) where \(D\) is a quadriculated disk. A 3-dimensional domino is the union of two adjacent unit cubes. Tilings by dominoes of 3-dimentional cubiculated regions are considered. A flip is a local move in tilings: two adjacent parallel dominoes are removed and placed back in a different position. Two tilings \(\mathfrak t_0\) and \(\mathfrak t_1\) are in the same connected component under flips if they can be joined by a finite sequence of flips. The twist associates to any tiling \(\mathfrak t\) of \(R_N\) an integer \(Tw(\mathfrak t)\). The twist is invariant under flips. A quadriculated disk \(D\) is regular if for any tilings \(\mathfrak t_0\), \(\mathfrak t_1\) of \(R_N\), the equality \(Tw(\mathfrak t_0) = Tw(\mathfrak t_1)\) implies that \(\mathfrak t_0\) and \(\mathfrak t_1\) can be joined by a sequence of flips provided some extra vertical space is allowed. In the paper, it is proved that for tilings of nontrivial cylinders \(R_N\) the twist follows the normal distribution when \(N\) goes to infinity. For regular disks, the author describes the larger connected components under flips of the set of tilings of \(R_N\). Also some more general results are obtained which use the domino group.