# The domino shuffling algorithm and Anisotropic KPZ stochastic growth

## TL;DR

This paper demonstrates that the domino shuffling algorithm models a stochastic growth process in the anisotropic KPZ class, revealing properties of growth speed, fluctuations, and discontinuities at specific slopes, using geometric and probabilistic methods.

## Contribution

It establishes the anisotropic KPZ classification for the domino shuffling growth model and analyzes the discontinuities and fluctuation behavior without explicit formulas.

## Key findings

- Growth speed depends on slope and weights, belonging to the anisotropic KPZ class.
- Height fluctuations grow at most logarithmically in time.
- Discontinuities in growth speed occur at finitely many smooth slopes.

## 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.