The domino shuffling algorithm and Anisotropic KPZ stochastic growth

Chhita, Sunil; Toninelli, Fabio

Annales Henri Lebesgue, Volume 4 (2021), pp. 1005-1034.

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 $v_{\mathtt{w}}\left(\rho \right)$ 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 [Ton18, Wol91]: one has $det\left[D^2{v}_{\mathtt{w}}\left(\rho \right)\right]<0$ and the height fluctuations grow at most logarithmically in time. Moreover, we prove that $D{v}_{\mathtt{w}}\left(\rho \right)$ 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 [CT19] via an explicit computation of $v_{\mathtt{w}}\left(\rho \right)$. 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.

L’algorithme de mélange de dominos (“domino shuffling algorithm”) [EKLP92a, EKLP92b, Pro03] peut être vu comme un processus stochastique qui décrit la croissance irréversible d’une interface discrète (2 + 1)-dimensionnelle [CT19, Zha18]. Sa vitesse stationnaire de croissance $v_{\mathtt{w}}\left(\rho \right)$ dépend de la pente moyenne $\rho$ de l’interface, aussi bien que des poids $\mathtt{w}$ des arêtes. On suppose que ces poids sont périodiques dans l’espace. Nous montrons que ce processus de croissance appartient à la classe KPZ anisotropique [Ton18, Wol91] : on a $det\left[D^2{v}_{\mathtt{w}}\left(\rho \right)\right]<0$ et les fluctuations de hauteur croissent au plus en logarithme du temps. De plus, nous montrons que $D{v}_{\mathtt{w}}\left(\rho \right)$ est discontinu à chacune des pentes « gazeuses » $\rho$ (il y en a un nombre fini). En correspondance avec ces pentes, les fluctuations ne divergent pas avec le temps. Pour un cas spécial de poids de périodicité $2$, des résultats analogues ont été montrés récemment [CT19] grâce à un calcul explicite de $v_{\mathtt{w}}\left(\rho \right)$. Dans le cas général, un tel calcul n’est pas faisable ; notre preuve passe plutôt par une relation entre la vitesse de croissance et la forme limite des pavages par dominos du diamant Aztèque.

Classification: 60K35, 82C41
Mots-clés : random tilings, stochastic interface growth, anisotropic KPZ

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