週期邊界條件下Temperley–Lieb哈密頓量於鏈結模上的馬可夫鏈研究

Abstract

本論文旨在研究一特定馬可夫鏈的收斂性質，此馬可夫鏈由 Temperley–Lieb 哈密頓量 $\\H_m = \\frac{1}{2m}\\sum_{j=1}^{2m} e_j$ 在非自交鏈結圖形 $\\LP(m)$ 上之作用所給出。此馬可夫鏈出現於可積 loop 模型與 Temperley–Lieb 代數的組合表示論等研究中。關於此鏈，一個重要結果是 Razumov–Stroganov（RS）對應，它將 $\\H_m$ 的基態與 fully-packed loop 模型上的均勻分布聯繫起來。 本論文分為兩大部分：首先，我們整理了 RS 對應及 Wieland 關於二面體對稱性（dihedral symmetry）的定理 [1, 2]，並呈現二面體對稱性的一個組合證明，這個證明能幫助我們更加理解本論文中馬可夫鏈的不變測度 $\\mu$。其次，藉由數值模擬，我們研究了不同初始條件下此鏈的收斂行為。具體而言，我們指出 $\\H_m$ 的譜隙數值上以 $m^{-1.9133}$ 的速度衰減，並發現了收斂速率基於奇偶性的二分現象。本論文最後討論了關於 cutoff 現象與 $m$ 足夠大時的漸近行為等開放問題。

We study the convergence properties of the Markov chain induced by the Temperley–Lieb Hamiltonian $\\H_m = \\frac{1}{2m}\\sum_{j=1}^{2m} e_j$ acting on the space of non-crossing link patterns $\\LP(m)$. This chain arises in the study of integrable loop models and combinatorial representations of the Temperley–Lieb algebra. One particularly important result regarding this chain is the Razumov–Stroganov (RS) correspondence, which connect the ground state of $\\H_m$ and the uniform distribution on fully-packed loop (FPL) configurations. This thesis is divided into two main parts. First, we survey the RS correspondence and Wieland's theorem on dihedral symmetry [1, 2]. A combinatorial proof of the dihedral symmetry is presented, which provides insight into understanding the invariant measure $\\mu$ of the Markov chain. Second, we numerically study the convergence behavior of the chain under different initial conditions. More precisely, we numerically show that the spectral gap of $\\H_m$ decays as $m^{-1.9133}$, and reveal a parity-dependent ``dichotomous" behavior for the convergence rates. Open questions regarding cutoff phenomena and large-$m$ asymptotics are discussed in the end of this paper.