在本篇文章中，我們將介紹在二維三角點陣上的簡單隨機漫步。我們
首先介紹位勢核函數a(x)，其中x ∈ Z2，我們求得在∥x∥ 趨近於無窮下，
a(x) 會近似於ln ∥x∥，並對其收斂速度進行討論。此外，假設Sn 為一在三角點陣上的簡單隨機漫步，我們觀察到a(Sn) 在不通過原點的情況下是為鞅，我們設Sn 的起始點位於大小兩圓B(R) 與B(r) 之間，利用可選停止定理，我們將a(·) 與逃脫兩圓之間機率做了連結，並且我們發現在R 趨近於無窮下先碰到大圓B(R) 的機率為O(1/ lnR)。在特別情況下，我們也能求得逃脫原點的機率。再者，比較三角點陣與正方點陣，我們觀察到兩者在逃脫大小圓的機率行為是沒有差別的。最後，我們介紹了有關調和測度與
容度，這些工具可以將我們的結果延伸至逃脫任意有限集合，我們也介紹
些定理證明調和測度是為從無窮遠處開始到入口點的機率，並一樣討論其
收斂速度。

In this thesis, we will introduce the simple random walk on the triangular lattice. We first introduce the potential kernel function a(x) for x ∈ Z2. We conclude that a(x) ≈ ln ∥x∥ as ∥x∥ → ∞. Moreover, the rate of convergence is
discussed too. Besides, let Sn be the simple random walk on the triangular lattice. We observe that a(Sn) is a
martingale without visiting the origin. We set our Sn
starting at the point between two circle, B(r) and B(R) with r < R. Using the optional stopping theorem, we make the connection between a(·) and escaping probability from two circle. Moreover, as R → ∞, we find that the probability
that visiting B(R) first is O(1/ lnR). In the specific case, we can also find the probability that escaping from the origin. Futhermore, compare triangular lattice with the square lattice, we observe that there is no difference between them in the behavior of escaping from circle.
Finally, we introduce the concept of harmonic measure and capacity. These can extend our results to calculate the probability of escaping from any finite set. We also introduce some theorem to prove that the harmonic measure is the probability of entrance point starting at infinity and also discuss the rate of convergence.

致謝 i
中文摘要 ii
Abstract iii
Contents iv
List of Tables v
List of Figures vi
1 Introduction 1
1.1 Random Walk 1
1.2 Triangular Lattice and Spread out Model 3
1.3 Notation 4
2 Main Result 5
2.1 Potential Kernel on Integer Lattice 5
2.2 Potential Kernel on Triangular Lattice 16
3 Proposition on Triangular Lattice 18
3.1 Escaping Probability 18
3.2 Green’s Function 25
4 Harmonic Measure and Capacity 27
4.1 Harmonic Measure 27
4.2 Capacity 31
Bibliography 33

[1] Robert Brown. Xxvii. a brief account of microscopical observations made in the months of june, july and august 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. The philosophical magazine, 4(21):161–173, 1828.
[2] Monroe D Donsker. An invariance principle for certain probability linit theorems. AMS, 1951.
[3] Albert Einstein et al. On the motion of small particles suspended in liquids at rest required by the molecularkinetic theory of heat. Annalen der physik, 17(549560): 208, 1905.
[4] Yasunari Fukai and Kôhei Uchiyama. Potential kernel for two dimensional random walk. The Annals of Probability, 24(4):1979–1992, 1996.
[5] Takashi Hara, Gordon Slade, and Remco van der Hofstad. Critical two point functions and the lace expansion for
spread out highdimensional percolation and related models. The Annals of Probability, 31(1):349–408, 2003.
[6] Gregory F Lawler and Vlada Limic. Random walk: a modern introduction, volume 123. Cambridge University Press, 2010.
[7] Paul Lévy. Propriétés asymptotiques des sommes de variables aléatoires indépendantes ou enchaînées. J. Math, 14(4), 1935.
[8] Karl Pearson. The problem of the random walk. Nature, 72(1867):342–342, 1905.
[9] Georg Pólya. Über eine aufgabe der wahrscheinlichkeitsrechnung betreffend die irrfahrt im straßennetz. Mathematische Annalen, 84(1):149–160, 1921.
[10] Serguei Popov. Two dimensional Random Walk: From Path Counting to Random Interlacements, volume 13. Cambridge University Press, 2021.
[11] Frank Spitzer. Principles of random walk, volume 34. Springer Science & Business Media, 2001.