Crossover distributions at the edge of the rarefaction fan
Crossover distributions at the edge of the rarefaction fan
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稀疏扇形边缘的交叉分布
DOI：
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发表时间：
2010
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通讯作者：
J. Quastel
中科院分区：
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作者：
Ivan Corwin;J. Quastel
本文研究了具有左漂移的简单排它过程的弱非对称极限，从步长Bernoulli初始数据出发，$\rho_-<\rho_+$使得宏观上有一个稀疏扇.我们研究了在适当的初始数据下，由Kardar-Parisi-Zhang（KPZ）方程的Hopf-科尔解给出的扇中沿沿着斜坡观测到的过程的波动。对于严格在扇内的斜率，初始数据是狄拉克δ函数，并且在[Comm. Pure Appl.Math.64（2011）466-537]和[Nuclear Phys. B 834（2010）523-542]中计算了单点分布函数。在稀疏扇的边缘，初始数据是片面的布朗。我们得到了一个新的交叉分布族，给出了该过程的精确一点分布，它以T\nearrow\infty$收敛于Airy $\mathcal{A}_{2\to\mathrm{BM}}$过程的交叉分布族.作为应用，我们证明了KPZ平衡态Hopf-Cole解的矩估计和大偏差估计。这些界限依赖于FKG不等式对随机热方程成立的明显的新观察。最后，通过Feynman-Kac路径积分，KPZ方程也控制着连续定向聚合物的自由能，因此我们的公式也可以用这些术语来解释。
We consider the weakly asymmetric limit of simple exclusion process with drift to the left, starting from step Bernoulli initial data with $\rho_-<\rho_+$ so that macroscopically one has a rarefaction fan. We study the fluctuations of the process observed along slopes in the fan, which are given by the Hopf--Cole solution of the Kardar-Parisi-Zhang (KPZ) equation, with appropriate initial data. For slopes strictly inside the fan, the initial data is a Dirac delta function and the one point distribution functions have been computed in [Comm. Pure Appl. Math. 64 (2011) 466-537] and [Nuclear Phys. B 834 (2010) 523-542]. At the edge of the rarefaction fan, the initial data is one-sided Brownian. We obtain a new family of crossover distributions giving the exact one-point distributions of this process, which converge, as $T\nearrow\infty$ to those of the Airy $\mathcal{A}_{2\to \mathrm{BM}}$ process. As an application, we prove moment and large deviation estimates for the equilibrium Hopf-Cole solution of KPZ. These bounds rely on the apparently new observation that the FKG inequality holds for the stochastic heat equation. Finally, via a Feynman-Kac path integral, the KPZ equation also governs the free energy of the continuum directed polymer, and thus our formula may also be interpreted in those terms.