  The domino-shuffling algorithm can be seen as a stochastic process describing
the irreversible growth of a $(2+1)$-dimensional discrete interface. Its
stationary speed of growth $v_{\mathtt w}(\rho)$ depends on the average
interface slope $\rho$, as well as on the edge weights $\mathtt w$, that are
assumed to be periodic in space. We show that this growth model belongs to the
Anisotropic KPZ class: one has $\det [D^2 v_{\mathtt w}(\rho)]<0$ and the
height fluctuations grow at most logarithmically in time. Moreover, we prove
that $D v_{\mathtt w}(\rho)$ is discontinuous at each of the (finitely many)
smooth (or "gaseous") slopes $\rho$; at these slopes, fluctuations do not
diverge as time grows. For a special case of spatially $2-$periodic weights,
analogous results have been recently proven in Chhita-Toninelli (2018) via an
explicit computation of $v_{\mathtt w}(\rho)$. In the general case, such a
computation is out of reach; instead, our proof goes through a relation between
the speed of growth and the limit shape of domino tilings of the Aztec diamond.
