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arXiv 1608.04726math.PRmath-phmath.MP
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稳态ASEP和六顶点模型的电流涨落

Current Fluctuations of the Stationary ASEP and Six-Vertex Model

Amol Aggarwal

备注：
83 pages, 13 figures. arXiv admin note: text overlap with arXiv:1607.08684. Version 2: Published version
发表信息：
Duke Math. J. 167, no. 2 (2018), 269-384
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摘要：

我们在本文中的结果是双重的。首先，我们考虑了稳态非对称简单排它过程（ASEP）的电流涨落，运行了很长一段时间的$T$，并表明它们是沿特征线的$T^{1 / 3}$阶。通过$T^{1 / 3}$定标，我们建立了这些波动收敛于固定KPZ方程的长时间高度波动，即Baik-Rains分布。这一结果在KPZ普适性的背景下早已被预测，特别是在该领域的许多结果的基础上，包括Ferrari和Spohn在2005年的工作（他们为TASEP建立了相同的结果），以及Balazs和Seppalainen在2010年的工作（他们为一般ASEP建立了$T^{1 / 3}$标度）。其次，我们介绍了一个任意的铁电体，对称六顶点模型的一个参数的斜坡家庭的特点，一类不变性吉布斯措施。这个斜率族对应于铁电六顶点模型的自由能分布中的所谓圆锥奇点（或三临界点）。我们认为波动的高度函数的模型上的一个大网格的大小$T$和显示，他们也是顺序$T^{1 / 3}$沿一定的特征线，这证实了预测Bukman和肖从1995年指出，铁电六顶点模型应表现出KPZ增长的圆锥形奇点。在通过$T^{1 / 3}$缩放高度波动后，我们再次在大的$T$极限中恢复Baik-Rains分布。根据（非对称）随机六顶点模型重新解释这一陈述，证实了Gwa和Spohn在1992年的预测。

英文摘要：

Our results in this paper are two-fold. First, we consider current fluctuations of the stationary asymmetric simple exclusion process (ASEP), run for some long time $T$, and show that they are of order $T^{1 / 3}$ along a characteristic line. Upon scaling by $T^{1 / 3}$, we establish that these fluctuations converge to the long-time height fluctuations of the stationary KPZ equation, that is, to the Baik-Rains distribution. This result has long been predicted under the context of KPZ universality and in particular extends upon a number of results in the field, including the work of Ferrari and Spohn in 2005 (who established the same result for the TASEP), and the work of Balazs and Seppalainen in 2010 (who established the $T^{1 / 3}$ scaling for the general ASEP). Second, we introduce a class of translation-invariant Gibbs measures that characterizes a one-parameter family of slopes for an arbitrary ferroelectric, symmetric six-vertex model. This family of slopes corresponds to what is known as the conical singularity (or tricritical point) in the free energy profile for the ferroelectric six-vertex model. We consider fluctuations of the height function of this model on a large grid of size $T$ and show that they too are of order $T^{1 / 3}$ along a certain characteristic line; this confirms a prediction of Bukman and Shore from 1995 stating that the ferroelectric six-vertex model should exhibit KPZ growth at the conical singularity. Upon scaling the height fluctuations by $T^{1 / 3}$, we again recover the Baik-Rains distribution in the large $T$ limit. Recasting this statement in terms of the (asymmetric) stochastic six-vertex model confirms a prediction of Gwa and Spohn from 1992.

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