平均过程的临界KPZ尺度
The critical KPZ scale for the Averaging Process
AI总结：
研究1+1维时空随机环境中随机游走模型的KPZ型极值涨落，通过平均过程发现一般矩准则不精确，实际KPZ极限在预测尺度外，证明基于多种定理和估计，还划分了模型不同临界区域。
AI中文摘要：
最近，在1 + 1维时空随机环境中的几种随机游走模型中证明了KPZ型极值涨落。一个一般的矩准则预测了这种行为应该出现的空间尺度，但它本身并不能保证在该尺度上有非平凡的涨落。本文表明，平均过程提供了一个该准则不精确的例子：由于更新机制中的简并性，实际的KPZ极限仅出现在预测尺度之外。相关的临界贡献来自两种不同涨落机制的相互作用，这一现象在随机游走环境设置中似乎相当特殊。我们的证明基于零和加性泛函的Dobrushin型局部时间极限定理、倾斜k点运动的精细估计以及最近基于矩的一维KPZ方程的Cole-Hopf解的公理特征。我们还确定了临界尺度两侧的行为，从而清晰地划分了模型的亚临界、临界和超临界区域。
英文摘要：
KPZ-type extremal fluctuations have recently been proved for several models of random walks in space-time random environments (RWRE) in $1+1$ dimensions. A general moment criterion predicts the spatial scale at which this behavior should occur, but does not by itself guarantee non-trivial fluctuations at that scale. In this paper, we show that the averaging process provides an instance in which this criterion is not sharp: because of a degeneracy in the update mechanism, the actual KPZ limit appears only beyond the predicted scale. The relevant critical contribution arises from the interplay of two distinct fluctuation mechanisms, a phenomenon that appears to be rather special within the RWRE setting. Our proof builds on Dobrushin-type local-time limit theorems for zero-sum additive functionals, refined estimates for tilted $k$-point motions, and the recent moment-based axiomatic characterization of Cole--Hopf solutions to the one-dimensional KPZ equation. We also identify the behavior on the two sides of the critical scale, thereby sharply separating the subcritical, critical, and supercritical regimes of the model.
