伯努利最終滲透模型之組合觀察

Abstract

最終滲透模型(last passage percolation model) 是一個在統計物理中重要的機率模型。其一些物件在特定隨機環境下可以藉由此模型的組合結構來確切求解。特別的是，Robinson-Schensted-Knuth (RSK) 對射高度連結了最終滲透值(last passage value) 和幾何分布的隨機環境。 這篇論文可以分成三個部分。首先，我們介紹了必需的組合工具，例如楊氏表(Young tableaux)、RSK 等等。其次，我們綜述了Baryshinokov 的研究工作[Bar01]，其使用了 RSK 獲得並以隨機矩陣Gaussian unitary ensemble (GUE) 來描述了在任意的隨機環境中歸一化的最終滲透值之極限分布。該證明的重要關鍵依靠特定的隨機環境：獨立同幾何分布(i.i.d. geometric distribution)，因其可以對應非負整數係數的矩陣，且這些矩陣可以經由RSK和最終滲透值連結。最後，我們研究了在獨立同伯努利分布下(i.i.d. Bernoulli distribution) 的最終滲透模型，該模型可由布林矩陣(binary matrices) 來描述。更精確地說，我們探索了於RSK對應中布林矩陣的結構，並證明了一些相關的組合結果。

The last passage percolation model (LPP) is an important probabilistic model in statistical mechanics. Some objects of the last passage percolation model are exactly solvable under some particular random environment through the combinatoric structures of the model. In particular, the Robinson–Schensted–Knuth (RSK) correspondence strongly connects the last passage value with the geometric random environment. The thesis is divided to three parts. Firstly, we introduce the required combinatoric tools, such as Young tableaux, RSK and so on. Secondly, we provide a survey of Baryshnikov's work [Bar01], which used RSK to verify the limiting distribution of the process of normalized last passage values under any random environment by the Gaussian unitary ensemble. The essential key to the proof depends on the specific random environment of i.i.d. geometric distribution, which can be identified by the matrices with non-negative integer entries, and the RSK correspondence gives a strong connection between such matrices and last passage values. Finally, we study the last passage percolation model under i.i.d. Bernoulli environment, which can be described by binary matrices. More precisely, we investigate the structure of binary matrices via the RSK correspondence, and show some combinatorial results.