The domino shuffling algorithm and Anisotropic KPZ stochastic growth
Annales Henri Lebesgue, Volume 4 (2021) , pp. 1005-1034.

Keywordsrandom tilings, stochastic interface growth, anisotropic KPZ

### Abstract

The domino-shuffling algorithm [EKLP92a, EKLP92b, Pro03] can be seen as a stochastic process describing the irreversible growth of a $\left(2+1\right)$-dimensional discrete interface [CT19, Zha18]. Its stationary speed of growth ${v}_{\mathtt{w}}\left(\rho \right)$ 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 [Ton18, Wol91]: one has $det\left[{D}^{2}{v}_{\mathtt{w}}\left(\rho \right)\right]<0$ and the height fluctuations grow at most logarithmically in time. Moreover, we prove that $D{v}_{\mathtt{w}}\left(\rho \right)$ 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 [CT19] via an explicit computation of ${v}_{\mathtt{w}}\left(\rho \right)$. 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.

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