探討最終滲流模型的橫向波動

Abstract

本論文探討在最終滲流模型（Last-Passage Percolation, LPP）中，最大路徑的橫向波動行為。LPP 模型描述的是在一個離散隨機環境中，從起點到終點所能取得的最大權重路徑。我們探討的焦點在於：當起終點距離與格點密度趨近無限時，最佳路徑偏離對角線的程度。 為了分析此現象，我們研究了特定 LPP 在極限下的行為，並證明其會收斂至帕松點過程（Poisson Point Process, PPP）。藉由建立 PPP 與最長遞增子序列模型（Longest Increasing Subsequence, LIS）之間的對應關係，我們引用 LIS 中已知的分布結果（包含 Tracy–Widom 分布），來分析 PPP 中路徑的橫向波動行為。我們闡明了 PPP 中的橫向波動指數為$\xi=2/3$，並解釋 LPP 在極限下同樣滿足此指數。

This thesis investigates the transversal fluctuations of maximal paths in the last-passage percolation (LPP) model. The LPP model describes the path of maximal weight in a discrete, random environment, typically represented as a lattice with i.i.d. random weights. We focus on understanding how much the optimal path deviates from the diagonal as the lattice becomes increasingly refined. To analyze this behavior, we study the scaling limit of specific LPP models and its convergence to the Poisson point process (PPP). By establishing a correspondence between PPP and the longest increasing subsequence (LIS) model, we employ known distributional results of LIS, including the Tracy–Widom distribution, to evaluate the transversal fluctuation behavior in PPP. We elucidate that the transversal fluctuation exponent of PPP is $\xi=2/3$, and demonstrate that LPP shares the same exponent in the scaling limit.