Domino tilings and flips in dimensions 4 and higher
Caroline J. Klivans, Nicolau C. Saldanha ยท Algebraic Combinatorics ยท 2022
In this paper we consider domino tilings of bounded regions in dimension n โฅ 4 . We define the twist of such a tiling, an elements of โค / ( 2 ) , and prove that it is invariant under flips, a simple local move in the space of tilings. We investigate which regions ๐ are regular , i.e. whenever two tilings t 0 and t 1 of ๐ ร [ 0 , N ] have the same twist then t 0 and t 1 can be joined by a sequence of flips provided some extra vertical space is allowed. We prove that all boxes are regular except ๐ = [ 0 , 2 ] 3 . Furthermore, given a regular region ๐ , we show that there exists a value M (depending only on ๐ ) such that if t 0 and t 1 are tilings of equal twist of ๐ ร [ 0 , N ] then the corresponding tilings can be joined by a finite sequence of flips in ๐ ร [ 0 , N + M ] . As a corollary we deduce that, for regular ๐ and large N , the set of tilings of ๐ ร [ 0 , N ] has two twin giant components under flips, one for each value of the twist.