动力学布朗末程渗透中的测地线迹线
Geodesic traces in dynamical Brownian last passage percolation
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中文总结 AI 辅助
本文研究动力学布朗末程渗透中测地线的并集访问单元数，通过稳定性估计与多尺度分析，证明其至多为$n^{1+\u03b5}$，并进一步得出非平凡双测地线时间集的Hausdorff测度为零。
中文摘要 AI 辅助
我们考虑布朗末程渗透（BLPP），其中每个单位水平区间上的布朗增量过程以速率1独立重新采样。通过将强通过时间稳定性估计与静态近最优路径的多尺度分析相结合，我们证明：对于每个$\u03b5>0$，在两个横向宽度为$n^{2/3}$阶、纵向长度为$n$阶、相距$n$阶的KPZ尺度矩形之间的所有测地线的并集，在临界时间区间$[0,n^{-1/3}]$内，在主体中访问的单位水平单元数至多为$n^{1+\u03b5}$，这一结论既在期望意义下成立，也以拉伸指数高概率成立。我们还获得了期望命中集大小的定量界$n\ueeCexp{C(\u03b4log\u03b4log n)^2}$，相应的失败概率至多为$Ce^{-c(\u03b4log n)^2}$。利用这一点，我们证明：对于次多项式衰减的规范$H(r)=\ueeCexp{-L(r)^2\u03b4log L(r)}$，其中$L(r)=\u03b4log\u03b4log(1/r)$，当$r\ueeCdownarrow0$时，存在非平凡双测地线的时间集几乎必然具有零Hausdorff测度。特别地，该集合几乎必然具有Hausdorff维数零。对于每个固定的确定性非轴向方向，我们进一步证明几乎必然没有时间在该方向上存在双测地线。
英文摘要
We consider Brownian last passage percolation (BLPP) in which the Brownian increment process on each unit horizontal interval is independently resampled at rate one. By combining strong passage-time stability estimates with a multiscale analysis of static near-optimal paths, we show that, for every $\varepsilon>0$, the union of all geodesics between two KPZ-scale rectangles of transverse width of order $n^{2/3}$ and longitudinal length of order $n$, separated by a distance of order $n$, visits at most $n^{1+\varepsilon}$ unit horizontal cells in the bulk during the critical time interval $[0,n^{-1/3}]$, both in expectation and with stretched-exponentially high probability. We also obtain the quantitative bound $n\exp\{C(\log\log n)^2\}$ on the expected hitset size, with a corresponding failure probability at most $Ce^{-c(\log n)^2}$. Using this, we establish that the set of times admitting a non-trivial bigeodesic has almost surely zero Hausdorff measure for the subpolynomially decaying gauge $H(r)=\exp\{-L(r)^2\log L(r)\}$, where $L(r)=\log\log(1/r)$, as $r\downarrow0$. In particular, this set almost surely has Hausdorff dimension zero. For each fixed deterministic non-axial direction, we further show that almost surely no time admits a bigeodesic in that direction.
发表机构
- University of Geneva（日内瓦大学）
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