论文标题
在管理Airy $ _1 $流程的相关衰减的指数上
On the exponent governing the correlation decay of the Airy$_1$ process
论文作者
论文摘要
我们研究了Airy $ _1 $过程的协方差的衰减，$ \ Mathcal {a} _1 $，这是$ \ Mathbb {r} $上的固定随机过程，它是Kardar-Parisi-parisi-Zhang（kpz）Universality classitaly classital classitaly classitaly classitaly classital in kardar-parisi-parisi-parisi-parisi-parisi-parisi-parisi-parising限制的限制。我们表明，衰减是超高的，并通过证明$ \ textrm {cov}（\ Mathcal {a} _1（0），\ Mathcal {a} _1（u）= e^e^{ - （\ frac {frac {4} {3}+O^33，该证明采用了概率技术和综合概率估计的组合。上限使用$ \ Mathcal {a} _1 $的连接到平面指数的最后一段段落，并在后一种模型中对点对上的大地测量学的几何形状的几个新结果具有独立的兴趣；尽管下限主要是分析性的，但使用Fredholm的决定词表达式，用于Airy $ _1 $过程的两个点函数以及FKG不等式。
We study the decay of the covariance of the Airy$_1$ process, $\mathcal{A}_1$, a stationary stochastic process on $\mathbb{R}$ that arises as a universal scaling limit in the Kardar-Parisi-Zhang (KPZ) universality class. We show that the decay is super-exponential and determine the leading order term in the exponent by showing that $\textrm{Cov}(\mathcal{A}_1(0),\mathcal{A}_1(u))= e^{-(\frac{4}{3}+o(1))u^3}$ as $u\to\infty$. The proof employs a combination of probabilistic techniques and integrable probability estimates. The upper bound uses the connection of $\mathcal{A}_1$ to planar exponential last passage percolation and several new results on the geometry of point-to-line geodesics in the latter model which are of independent interest; while the lower bound is primarily analytic, using the Fredholm determinant expressions for the two point function of the Airy$_1$ process together with the FKG inequality.
