Almost sure bounds for higher-order derivatives of first-passage percolation with respect to the environment
Ivan Matic, Rados Radoicic, Dan Stefanica
公開日: 2025/4/11
Abstract
The variance of the first-passage percolation is bounded by the $L^2$-norms of the derivatives with respect to the environment. In this paper we prove that the environment derivatives of orders $k\in\{2,3,4\}$ are bounded below by $-\binom{k-2}{\lceil\frac{k-2}2\rceil}$ and above by $\binom{k-2}{\lceil\frac{k-2}2\rceil}$. We believe that these are the bounds for all $k$. We provide examples of environments that show that these extreme values can be attained.