The domino shuffling algorithm and Anisotropic KPZ stochastic growth

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