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We consider domino tilings of $3$-dimensional cubiculated regions. Athree-dimensional domino is a 2x2x1 rectangular cuboid. We are particularlyinterested in regions of the form $R_N = D \times [0,N]$ where $D$ is a fixedquadriculated disk. In dimension 3, the twist associates to each tiling $t$ aninteger $Tw(t)$. We prove that, when $N$ goes to infinity, the twist follows anormal distribution.  A flip is a local move: two neighboring parallel dominoes are removed andplaced back in a different position. The twist is invariant under flips. Aquadriculated disk $D$ is regular if, whenever two tilings $t_0$ and $t_1$ of$R_N$ satisfy $Tw(t_0) = Tw(t_1)$, $t_0$ and $t_1$ can be joined by a sequenceof flips provided some extra vertical space is allowed. Many large disks areregular, including rectangles $D = [0,L] \times [0,M]$ with $LM$ even and $L,M\ge 3$. For regular disks, we describe the larger connected components underflips of the set of tilings of the region $R_N = D \times [0,N]$. As acorollary, let $p_N$ be the probability that two random tilings $T_0$ and $T_1$of $D \times [0,N]$ can be joined by a sequence of flips conditional to theirtwists being equal. Then $p_N$ tends to 1 if and only if $D$ is regular.  Under a suitable equivalence relation, the set of tilings has a groupstructure, the domino group. These results illustrate the fact that the dominogroup dictates many properties of the space of tilings of the cylinder $R_N = D\times [0,N]$, particularly for large $N$.

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