The domino shuffling algorithm and Anisotropic KPZ stochastic growth.

Abstract

The domino-shuffling algorithm [EKLP92a, EKLP92b, Pro03] can be seen as a stochastic process describing the irreversible growth of a (2+1)-dimensional discrete interface [CT19, Zha18]. Its stationary speed of growth 𝑣𝚠(𝜌) depends on the average interface slope 𝜌, as well as on the edge weights 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[𝐷2𝑣𝚠(𝜌)]<0 and the height fluctuations grow at most logarithmically in time. Moreover, we prove that 𝐷𝑣𝚠(𝜌) is discontinuous at each of the (finitely many) smooth (or “gaseous”) slopes 𝜌; 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 𝑣𝚠(𝜌). 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.