标题： 首次通过渗流中的中偏差对于有界权重 显示英文标题

标题： Moderate deviations in first-passage percolation for bounded weights

摘要： We investigate the moderate and large deviations in first-passage percolation (FPP) with bounded weights on $\mathbb{Z}^d$ for $d \geq 2$. Write $T(\mathbf{x}, \mathbf{y})$ for the first-passage time and denote by $\mu(\mathbf{u})$ the time constant in direction $\mathbf{u}$. 在本文中，我们证明了，如果假设次线性误差项$T(\mathbf{0}, N\mathbf{u}) - N\mu(\mathbf{u})$的阶为$N^\chi$，则在一些未经验证（但广受相信）的假设下，对于$\chi < a < 1$，\begin{align*} &\mathbb{P}\bigl(T(\mathbf{0}, N\mathbf{u}) > N\mu(\mathbf{u}) + N^a\bigr) = \exp{\Big(-\,N^{\frac{d(1+o(1))}{1-\chi}(a-\chi)}\Big)}, &\mathbb{P}\bigl(T(\mathbf{0}, N\mathbf{u}) < N\mu(\mathbf{u}) - N^a\bigr) = \exp{\Big(-\,N^{\frac{1+o(1)}{1-\chi}(a-\chi)}\Big)}, \end{align*}在临界情况下伴有估计值$a=1$。 此外，指数$\frac{d}{1-\chi}$和$\frac{1}{1-\chi}$也出现在上尾和下尾大偏差率函数在$0$附近的渐近行为中。值得注意的是，我们的一些估计是严格建立的，而无需依赖任何未经验证的假设。我们的主要结果突显了波动与大偏差衰减率之间的相互作用，并弥合了这两个区域之间的差距。我们证明的一个关键要素是通过多尺度分析对若干中偏差估计进行了改进的集中性分析，这一现象之前曾在二维最后一段渗流和二维旋转不变FPP的背景下出现过。 显示英文摘要

摘要： We investigate the moderate and large deviations in first-passage percolation (FPP) with bounded weights on $\mathbb{Z}^d$ for $d \geq 2$. Write $T(\mathbf{x}, \mathbf{y})$ for the first-passage time and denote by $\mu(\mathbf{u})$ the time constant in direction $\mathbf{u}$. In this paper, we establish that, if one assumes that the sublinear error term $T(\mathbf{0}, N\mathbf{u}) - N\mu(\mathbf{u})$ is of order $N^\chi$, then under some unverified (but widely believed) assumptions, for $\chi < a < 1$, \begin{align*} &\mathbb{P}\bigl(T(\mathbf{0}, N\mathbf{u}) > N\mu(\mathbf{u}) + N^a\bigr) = \exp{\Big(-\,N^{\frac{d(1+o(1))}{1-\chi}(a-\chi)}\Big)}, &\mathbb{P}\bigl(T(\mathbf{0}, N\mathbf{u}) < N\mu(\mathbf{u}) - N^a\bigr) = \exp{\Big(-\,N^{\frac{1+o(1)}{1-\chi}(a-\chi)}\Big)}, \end{align*} with accompanying estimates in the borderline case $a=1$. Moreover, the exponents $\frac{d}{1-\chi}$ and $\frac{1}{1-\chi}$ also appear in the asymptotic behavior near $0$ of the rate functions for upper and lower tail large deviations. Notably, some of our estimates are established rigorously without relying on any unverified assumptions. Our main results highlight the interplay between fluctuations and the decay rates of large deviations, and bridge the gap between these two regimes. A key ingredient of our proof is an improved concentration via multi-scale analysis for several moderate deviation estimates, a phenomenon that has previously appeared in the contexts of two-dimensional last-passage percolation and two-dimensional rotationally invariant FPP.