2022/09/07 by Raphael de Marreiros, de Marreiros, Raphael
Computer Science · Materials Science · Mathematics · #05C70 #Cellular Automata and Applications #Combinatorics (math.CO) #FOS: Mathematics #Mathematical Dynamics and Fractals #Primary 05B45 #Quasicrystal Structures and Properties #Secondary 52C22
paper · pdf · doi:10.48550/arxiv.2209.03109
openalex publication_date 2022/09/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider three-dimensional domino tilings of cylinders RN = D × [0,N] where D ⊂ ℝ2 is a fixed quadriculated disk and N ∈ ℕ. A domino is a 2 × 1 × 1 brick. A flip is a local move in the space of tilings T(RN): remove two adjacent dominoes and place them back after a rotation. The twist is a flip invariant which associates an integer number to each tiling. For some disks D, called regular, two tilings of RN with the same twist can be joined by a sequence of flips once we add vertical space to the cylinder. We have that if D is regular then the size of the largest connected component under flips of T(RN) is Θ(N-(1)/(2)|T(RN)|). The domino group GD captures information of the space of tilings. A disk D is regular if and only if GD is isomorphic to ℤ ⊕ ℤ/(2); sufficiently large rectangles are regular. We prove that certain families of disks are irregular. We show that the existence of a bottleneck in a disk D often implies irregularity. In many, but not all, of these cases, we also prove that D is strongly irregular, i.e., that there exists a surjective homomorphism from GD+ (a subgroup of index two of GD) to the free group of rank two. Moreover, we show that if D is strongly irregular then the cardinality of the largest connected component under flips of T(RN) is O(cN |T(RN)|) for some c ∈ (0,1).