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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.
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https://hal.archives-ouvertes.fr/hal-02159744
Contributor : Fabio Toninelli <>
Submitted on : Wednesday, June 19, 2019 - 7:43:54 AM
Last modification on : Friday, January 10, 2020 - 11:20:03 AM

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  • HAL Id : hal-02159744, version 1
  • ARXIV : 1906.07231

Citation

Sunil Chhita, Fabio Toninelli. The domino shuffling algorithm and Anisotropic KPZ stochastic growth. 2019. ⟨hal-02159744⟩

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