论文标题
最后一段渗透中的网格熵 - 一种超级临界指数方法
Grid entropy in last passage percolation -- a superadditive critical exponent approach
论文作者
论文摘要
在I.I.D.的环境中工作在$ \ mathbb {r}^d $上进行的最后一个通用渗透，没有对基础边缘\ hyp {}重量分布的假设，我们得出了网格熵的概念 - 一种亚无良性定理的概述，与经验量的阶梯属性相关的阶段较弱或等效地构成了确定性的阶段，确定性地构成了确定性的阶段， Levy-Prokhorov公制。这为Rassoul-Agha和Seppäläinen首先开发的熵提供了一种新的方法，作为沿路径经验措施的较大偏差率函数。在其2014年论文ARXIV：1202.2584中，作为该熵的凸结合物，为点对点/点对上的吉布斯自由能开发了变异公式。我们在新框架中重新制作这些公式，并明确将我们对网格熵的描述与它们的描述联系起来。我们还通过在不等式中引入相对熵项来改善该熵的已知结合。此外，我们表明，具有有限网格熵的一组度量与贝茨最近的论文ARXIV：2006.12580在BATES中研究的经验度量的确定性限制点相吻合。此外，我们部分回答了Hoffman问题的定向聚合物版本，该问题以前是Bates在零温度案例中解决的。我们的结果涵盖了点对点和点对上的场景。
Working in the setting of i.i.d. last-passage percolation on $\mathbb{R}^D$ with no assumptions on the underlying edge\hyp{}weight distribution, we arrive at the notion of grid entropy - a Subadditive Ergodic Theorem limit of the entropies of paths with empirical measures weakly converging to a given target, or equivalently a deterministic critical exponent of canonical order statistics associated with the Levy-Prokhorov metric. This provides a fresh approach to an entropy first developed by Rassoul-Agha and Seppäläinen as a large deviation rate function of empirical measures along paths. In their 2014 paper arXiv:1202.2584, variational formulas are developed for the point-to-point/point-to-level Gibbs Free Energies as the convex conjugates of this entropy. We rework these formulas in our new framework and explicitly link our descriptions of grid entropy to theirs. We also improve on a known bound for this entropy by introducing a relative entropy term in the inequality. Furthermore, we show that the set of measures with finite grid entropy coincides with the deterministic set of limit points of empirical measures studied in a recent paper arXiv:2006.12580 by Bates. In addition, we partially answer a directed polymer version of a question of Hoffman which was previously tackled in the zero temperature case by Bates. Our results cover both the point-to-point and point-to-level scenarios.
