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    政大機構典藏 > 理學院 > 應用數學系 > 學位論文 >  Item 140.119/146300
    Please use this identifier to cite or link to this item: https://nccur.lib.nccu.edu.tw/handle/140.119/146300


    Title: 兩個獨立簡單隨機漫步在 d 維整數晶格之臨界行為
    The critical behavior of two independent simple random walks on Z^d lattices
    Authors: 李柏駿
    Li, Bo-Jyun
    Contributors: 陳隆奇
    Chen, Lung-Chi
    李柏駿
    Li, Bo-Jyun
    Keywords: 簡單隨機漫步
    懶惰隨機漫步
    格林函數
    位勢核
    逃脫機率
    選擇停止定理
    Simple random walk
    The lazy random walk
    Green’s function
    Potential kernel
    Escape probability
    Optional stopping theorem
    Date: 2023
    Issue Date: 2023-08-02 13:02:39 (UTC+8)
    Abstract: 在這篇論文中,我們介紹兩個獨立的簡單隨機漫步在 d 維整數晶
    格上的臨界行為。假設這兩個隨機漫步的起點分別為 a 和 b,且滿足
    2r < |a − b| < 2R 的條件。我們將研究在每個維度上,兩個獨立的簡單隨機漫步在距離小於 2r 之前距離超過 2R 的機率。也就是說,這是兩個獨立簡單隨機漫步的「逃脫機率」。
    為了解決這個問題,我們首先介紹了懶惰隨機漫步 Ln,它由兩個獨
    立的簡單隨機漫步的相對位置生成。我們討論了懶惰隨機漫步的格林函
    數 G(x),其中 x ∈ Zd 且 d ≥ 3,以及懶惰隨機漫步的勢能核 a(x),其中x ∈ Z2。我們將展示當 |x| 趨於無窮並且位於偶數位置時,G(x) 與簡單隨機漫步的格林函數相同並以相同的速度收斂。同樣地,a(x) 與簡單隨機漫步的勢能核相同並以相同的速度收斂。基於此,我們觀察到 G(Ln) 和 a(Ln)在沒有碰到原點的情況是鞅。通過應用選擇停止定理,我們建立了格林函數、勢能核和逃脫機率之間的聯繫。
    此外,我們也會探討兩個獨立簡單隨機漫步路徑的交錯次數的期望值,
    我們求得在維度 d ≥ 5 的整數晶格上,兩個獨立簡單隨機漫步的路徑交錯有限多次的機率等於 1;在維度 d ≤ 4 的整數晶格上,兩個獨立簡單隨機漫步的路徑交錯無窮多次的機率等於 1。
    In this thesis, we introduce the critical behavior of two independent simple random walks on the Zd lattices. We suppose that the starting points of these two random walks are denoted by a and b, respectively, and satisfy the condition 2r <|a−b| < 2R. Our objective is to investigate the probability of the distance between these two independent simple random walks exceeding 2R before it becomes less than 2r in any dimensional lattice. This probability is commonly referred to as the ”escape probability” of two independent simple random walks.
    To address this question, we first introduce the lazy random walk Ln, which is generated by the relative positions of two independent simple random walks. We delve into the discussion of the Green’s function G(x) of the lazy random walk, where x ∈ Zd and d ≥ 3, as well as the potential kernel a(x) of the lazy random walk, where x ∈ Z2. We aim to demonstrate that, as the magnitude of |x| tends to infinity and x lies on an even site, G(x) is equivalent to the Green’s function of the simple random walk and converges at the same rate. Likewise, a(x) is equivalent to the potential kernel of the simple random walk and converges at the same rate. Drawing from this, we observe that G(Ln) and a(Ln) act as martingales when they avoid reaching the origin. By applying the optional stopping theorem, we establish a connection between the Green’s function, potential kernel, and the escape probability.
    Furthermore, we explore the expected number of intersections between the paths of two independent simple random walks. We establish a proof on the Zd lattice, where d ≥ 5, the probability of the paths intersecting for a finite number
    of times is equal to 1. On the Zd lattice, where d ≤ 4, the probability of the paths intersecting for an infinite number of times is equal to 1.
    Reference: [1] Robert Brown. Brownian motion. Unpublished experiment, 38, 1827.
    [2] Edward A Codling, Michael J Plank, and Simon Benhamou. Random walk models in
    biology. Journal of the Royal society interface, 5(25):813–834, 2008.
    [3] Yasunari Fukai and Kôhei Uchiyama. Potential kernel for two-dimensional random walk.
    The Annals of Probability, 24(4):1979–1992, 1996.
    [4] Philippe Marchal. Constructing a sequence of random walks strongly converging to
    brownian motion. Discrete Mathematics & Theoretical Computer Science, 2003.
    [5] Karl Pearson. The problem of the random walk. Nature, 72(1865):294–294, 1905.
    [6] Georg Pólya. Über eine aufgabe der wahrscheinlichkeitsrechnung betreffend die irrfahrt
    im straßennetz. Mathematische Annalen, 84(1-2):149–160, 1921.
    [7] Serguei Popov. Two-Dimensional Random Walk: From Path Counting to Random
    Interlacements, volume 13. Cambridge University Press, 2021.
    [8] Enrico Scalas. The application of continuous-time random walks in finance and economics.
    Physica A: Statistical Mechanics and its Applications, 362(2):225–239, 2006.
    [9] Frank Spitzer. Principles of random walk, volume 34. Springer Science & Business Media,
    2013.
    [10] Kôhei Uchiyama. Green’s functions for random walks on ℤn. Proceedings of the London
    Mathematical Society, 77(1):215–240, 1998.
    [11] George H Weiss and Robert J Rubin. Random walks: theory and selected applications.
    Advances in Chemical Physics, 52:363–505, 1983.
    Description: 碩士
    國立政治大學
    應用數學系
    110751001
    Source URI: http://thesis.lib.nccu.edu.tw/record/#G0110751001
    Data Type: thesis
    Appears in Collections:[應用數學系] 學位論文

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