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.
Highlights
In the realm of stochastic interface growth [BS95], dimension (2+1) plays a distinguished role
The universality class of a model is determined by the properties of the average speed of growth v(ρ) limt → ∞
Some of these results have been extended to an AKPZ growth process defined in terms of the dimer model on the square grid, see [CFT19]
Summary
In the realm of stochastic interface growth [BS95], dimension (2+1) (i.e., growth of a two-dimensional interface in three-dimensional physical space) plays a distinguished role. This is a discrete-time Markov chain on perfect matchings (or “domino tilings”) of Z2, that was originally devised [EKLP92a, EKLP92b, Pro03] as a way to exactly sample and to count perfect matchings of certain special two-dimensional domains (Aztec diamonds) When this algorithm is run on the infinite square grid, it can be seen as a (2 + 1)-dimensional growth model, and it is from this point of view that we consider it here. Our main result is that the domino shuffling algorithm (with general weights w) belongs to the AKPZ class, and that the speed of growth is singular at each of the smooth slopes (see Theorem 2.3 and Section 2.4.1 for more precise statements): Main Theorem (Informal version). The result of [Zha18] is stated for the case of edge weights with space periodicity 1, but the same proof presumably works for general periodic edge weights, as in the framework of the present article
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