The domino shuffling algorithm and Anisotropic KPZ stochastic growth

Abstract

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.

30 pages 9 figures; v3: minor changes

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Cited by in corpus (4)

• The stationary AKPZ equation: logarithmic superdiffusivity
• On the domino shuffle and matrix refactorizations
• Universality of deterministic KPZ
• 2D Anisotropic KPZ at stationarity: scaling, tightness and non triviality