Brownian structure in the KPZ fixed point
Brownian structure in the KPZ fixed point
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DOI：
10.24033/ast.1200
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发表时间：
2019-12
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影响因子：
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通讯作者：
Jacob Calvert;A. Hammond;Milind Hegde
中科院分区：
文献类型：
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作者：
Jacob Calvert;A. Hammond;Milind Hegde
一维局域随机增长的许多模型被认为属于Kardar-Parisi-Zhang(KPZ)普适类。对于这样的模型，可以在通过三分之一和三分之二的特征KPZ标度指数指定的标度坐标中查看提前时间的界面轮廓。当取这个定标界面的长时间限制时，预期--并证明了一些可积模型--直到抛物线移位，获得了艾里$2$进程$\mathcal{A}：\mathbb{R}\to\mathbb{R}$。这一过程可以通过Robinson-Schensted-Knuth对应嵌入，作为随机连续曲线的$\mathbb{N}$索引系统中的最上面的曲线，即艾里线系综。我们的主要结果之一是断言：Aary$2$过程与单位阶区间上的布朗运动$B$有很强的相似性，因此，关于这个区间上的$B$的定律的Radon-Nikodym导数存在于$p的每个$L^p$空间中.我们的证明技术利用了抛物线平移后的艾里线系综所满足的概率重采样或{em Brown Gibbs}性质，本文发展了从[CH14]开始并在[Ham19a]中继续进行的这一系综的布朗Gibbs分析。我们对比例界面轮廓的布朗比较是正在进行的通过概率和几何证明方法研究KPZ普适性的方案的一个元素，辅以有限但必要的可积输入使用。事实上，比较结果是研究这一普适性类的有用工具。我们提出并证明了几个应用，例如布朗最后一程渗流中的近基态结构，或从一类非常一般的初始数据中的任何元素演化而来的标度界面轮廓中的布朗结构。
Many models of one-dimensional local random growth are expected to lie in the Kardar-Parisi-Zhang (KPZ) universality class. For such a model, the interface profile at advanced time may be viewed in scaled coordinates specified via characteristic KPZ scaling exponents of one-third and two-thirds. When the long time limit of this scaled interface is taken, it is expected -- and proved for a few integrable models -- that, up to a parabolic shift, the Airy$_2$ process $\mathcal{A}:\mathbb{R} \to \mathbb{R}$ is obtained. This process may be embedded via the Robinson-Schensted-Knuth correspondence as the uppermost curve in an $\mathbb{N}$-indexed system of random continuous curves, the Airy line ensemble. Among our principal results is the assertion that the Airy$_2$ process enjoys a very strong similarity to Brownian motion $B$ (of rate two) on unit-order intervals; as a consequence, the Radon-Nikodym derivative of the law of $\mathcal{A}$ on say $[-1,1]$, with respect to the law of $B$ on this interval, lies in every $L^p$ space for $p \in (1,\infty)$. Our technique of proof harnesses a probabilistic resampling or {\em Brownian Gibbs} property satisfied by the Airy line ensemble after parabolic shift, and this article develops Brownian Gibbs analysis of this ensemble begun in [CH14] and pursued in [Ham19a]. Our Brownian comparison for scaled interface profiles is an element in the ongoing programme of studying KPZ universality via probabilistic and geometric methods of proof, aided by limited but essential use of integrable inputs. Indeed, the comparison result is a useful tool for studying this universality class. We present and prove several applications, concerning for example the structure of near ground states in Brownian last passage percolation, or Brownian structure in scaled interface profiles that arise from evolution from any element in a very general class of initial data.