A (2+1)-dimensional growth process with explicit stationary measures
A (2+1)-dimensional growth process with explicit stationary measures
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具有显式平稳测量的 (2 1) 维增长过程
DOI：
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发表时间：
2015
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通讯作者：
F. Toninelli
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作者：
F. Toninelli
引入了一类（2+1）维随机增长过程，它可以看作是离散界面的不可逆随机动力学。“不可逆”意味着界面具有平均非零漂移。界面构型对应于无限六边形或正方形晶格的二聚体覆盖的高度函数。该模型也可以被看作是一个相互作用的驱动粒子系统，在完全不对称的情况下，动态对应于一个相互作用的Hammersley过程的无限集合。
当动力学不对称参数$（p-q）$等于零时，无限体积吉布斯测度$pi_
ho$（给定斜率$
HO$）是固定的和可逆的。当$p
e q$，$pi_
但是，值得注意的是，它们仍然是静止的。在这样的定态中，我们发现在任意给定点x$处的平均高度函数随时间t$以非零速度线性增长：$mathbb E Q_x（t）：=mathbb E（h_x（t）-h_x（0））= V（
而$Q_x（t）$的典型波动小于$t$的任何幂，如$t 000美元。
在p=0，q=1的完全非对称情形下，在六方晶格上，动力学符合A.鲍罗丁和P.L.法拉利对于一个合适的选择，“可积”，初始条件（这是非常远离稳态），他们能够确定的流体动力学限制和CLT的界面波动规模$sqrt{log t}$，利用这一事实，在这种情况下，某些时空高度相关性可以精确计算。
We introduce a class of (2+1)-dimensional stochastic growth processes, that can be seen as irreversible random dynamics of discrete interfaces. "Irreversible" means that the interface has an average non-zero drift. Interface configurations correspond to height functions of dimer coverings of the infinite hexagonal or square lattice. The model can also be viewed as an interacting driven particle system and in the totally asymmetric case the dynamics corresponds to an infinite collection of mutually interacting Hammersley processes.
When the dynamical asymmetry parameter $(p-q)$ equals zero, the infinite-volume Gibbs measures $pi_
ho$ (with given slope $
ho$) are stationary and reversible. When $p
e q$, $pi_
ho$ are not reversible any more but, remarkably, they are still stationary. In such stationary states, we find that the average height function at any given point $x$ grows linearly with time $t$ with a non-zero speed: $mathbb E Q_x(t):=mathbb E(h_x(t)-h_x(0))= V(
ho) t$ while the typical fluctuations of $Q_x(t)$ are smaller than any power of $t$ as $t oinfty$.
In the totally asymmetric case of $p=0,q=1$ and on the hexagonal lattice, the dynamics coincides with the "anisotropic KPZ growth model" introduced by A. Borodin and P. L. Ferrari. For a suitably chosen, "integrable", initial condition (that is very far from the stationary state), they were able to determine the hydrodynamic limit and a CLT for interface fluctuations on scale $sqrt{log t}$, exploiting the fact that in that case certain space-time height correlations can be computed exactly.