Stochastic Heat Equation Limit of a (2 + 1)d Growth Model
Stochastic Heat Equation Limit of a (2 + 1)d Growth Model
复制标题
(2 + 1)d 增长模型的随机热方程极限
DOI：
10.1007/s00220-016-2718-4
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发表时间：
2016
影响因子：
2.4
通讯作者：
F. Toninelli
中科院分区：
文献类型：
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作者：
A. Borodin;Ivan Corwin;F. Toninelli
我们确定一个$${q \to 1}$$q→1限制的二维q-惠特克驱动粒子系统的环面研究以前在Corwin和Toninelli（电子。可能吧21（44）：1-12，2016）。这有一个解释为（2 + 1）维随机界面生长模型，这被认为是属于所谓的各向异性Kardar-Parisi-Zhang（KPZ）类。这个限制福尔斯落入一个一般类的二维系统的驱动线性SDES有固定措施的梯度。以粒子数为无穷大，我们证明高斯自由场型波动的固定措施。考虑到平稳测度的时间演化，我们确定了沿着特征，相关性由（2 + 1）维可加随机热方程的相关性渐近给出.这证实了（对于该模型）预测，即（2 + 1）维中各向异性KPZ方程的非线性是不相关的。
We determine a $${q \to 1}$$q→1 limit of the two-dimensional q-Whittaker driven particle system on the torus studied previously in Corwin and Toninelli (Electron. Commun. Probab. 21(44):1–12, 2016). This has an interpretation as a (2 + 1)-dimensional stochastic interface growth model, which is believed to belong to the so-called anisotropic Kardar–Parisi–Zhang (KPZ) class. This limit falls into a general class of two-dimensional systems of driven linear SDEs which have stationary measures on gradients. Taking the number of particles to infinity we demonstrate Gaussian free field type fluctuations for the stationary measure. Considering the temporal evolution of the stationary measure, we determine that along characteristics, correlations are asymptotically given by those of the (2 + 1)-dimensional additive stochastic heat equation. This confirms (for this model) the prediction that the non-linearity for the anisotropic KPZ equation in (2 + 1)-dimension is irrelevant.