三角點陣上的簡單隨機漫步

Abstract

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\n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，\na(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\n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\n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\n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\ndiscussed too. Besides, let Sn be the simple random walk on the triangular lattice. We observe that a(Sn) is a\nmartingale without visiting the origin. We set our Sn\nstarting 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\nthat 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.\nFinally, 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.