高維度初抵滲透模型的時間常數與極限形狀

Abstract

初抵滲透模型是一個由 Hammersley 跟 Welsh 在 1965 年提出的廣為人知的統計物理模型。模型的定義如下。令$\mathbb{Z}^d$為$d$維的整數網格、$\mathcal{E}^d$為收集所有在$\mathbb{Z}^d$上的（無向）最近鄰邊的集合，也就是\[\mathcal{E}^d = \{\{\mathbf{x}, \mathbf{y}\} \mid \mathbf{x}, \mathbf{y} \in \mathbb{Z}^d, \|\mathbf{x} - \mathbf{y}\|_1 = 1\}.\]對於每條邊$e \in \mathcal{E}^d$，我們指定一個非負的隨機變數$\tau_e$, 稱為$e$的\textbf{權重（穿越時間）}。對於在$\mathbb{Z}^d$上任意的網格路徑$\gamma$，\textbf{$\gamma$的穿越時間}有如下的定義\[\tau(\gamma) = \sum_{e \in \gamma} \tau_e.\]對於兩個整數點$\mathbf{x}, \mathbf{y} \in \mathbb{Z}^d$，我們定義$\Gamma(\mathbf{x}, \mathbf{y})$為在$\mathbb{Z}^d$上所有連接$\mathbf{x}$與$\mathbf{y}$的網格路徑。對於一個點$\mathbf{x} \in \mathbb{R}^d$，我們定義$\left\lfloor\mathbf{x} \right\rfloor \in \mathbb{Z}^d$為唯一使得$\mathbf{x} \in \left\lfloor\mathbf{x}\right\rfloor + [0, 1)^d$的整數點。現在，我們定義\textbf{從$\mathbf{x}$到$\mathbf{y}$的初抵時間}為\[\tau(\mathbf{x}, \mathbf{y}) = \inf\{\tau(\gamma) \mid \gamma \in \Gamma(\left\lfloor\mathbf{x}\right\rfloor, \left\lfloor\mathbf{y}\right\rfloor)\}.\]

儘管模型本身很易於理解，但時至今日，仍有許多關於這個模型的問題懸而未決。其中一個就是關於時間常數與極限形狀的決定。在這篇碩士論文中，我們會考慮時間常數在維度趨向無限時的漸進行為，並以此在維度夠高時去否決部分的極限形狀候選。

First-passage percolation (FPP) is a widely recognized statistical physics model that was originally proposed by Hammersley and Welsh in 1965. The model is defined as follows. Let $\mathbb{Z}^d$ be the $d$-dimensional integer lattice and $\mathcal{E}^d$ be the collection of (non-oriented) nearest-neighbour edges in $\mathbb{Z}^d$. That is, \[\mathcal{E}^d = \{\{\mathbf{x}, \mathbf{y}\} \mid \mathbf{x}, \mathbf{y} \in \mathbb{Z}^d, \|\mathbf{x} - \mathbf{y}\|_1 = 1\}.\]We assign to each edge $e \in \mathcal{E}^d$ a non-negative random variable $\tau_e$, called the \textbf{weight (passage time)} of $e$. For any lattice path $\gamma$ on $\mathbb{Z}^d$, the \textbf{passage time of $\gamma$} is defined by \[\tau(\gamma) = \sum_{e \in \gamma} \tau_e.\]For two points $\mathbf{x}, \mathbf{y} \in \mathbb{Z}^d$, we define $\Gamma(\mathbf{x}, \mathbf{y})$ to be all lattice paths on $\mathbb{Z}^d$ connecting $\mathbf{x}$ to $\mathbf{y}$. For a point $\mathbf{x} \in \mathbb{R}^d$, we define the ``floor" of $\mathbf{x}$, $\left\lfloor\mathbf{x} \right\rfloor \in \mathbb{Z}^d$, to be the unique vertex in $\mathbb{Z}^d$ such that $\mathbf{x} \in \left\lfloor\mathbf{x}\right\rfloor + [0, 1)^d$. Now, we define the \textbf{first-passage time from $\mathbf{x}$ to $\mathbf{y}$} by\[\tau(\mathbf{x}, \mathbf{y}) = \inf\{\tau(\gamma) \mid \gamma \in \Gamma(\left\lfloor\mathbf{x}\right\rfloor, \left\lfloor\mathbf{y}\right\rfloor)\}.\]

Despite its apparent simplicity, numerous unresolved problems persist within this model. Among these, determining the time constant and the limit shape are particularly noteworthy. In this master thesis, we aim to investigate the asymptotic behavior of the time constant as the dimension tends to infinity and subsequently reject several potential limit shapes in sufficiently high dimensions.