邊界凍結滲流模型及其變體之探討

Abstract

我們研究了 Makowiec 在其碩士論文 [7] 中提出的邊界凍結滲流模型，該模型定義在三角格子的有限子圖 B(N) 上，其中 B(N) 表示以原點為中心、邊長為 2N的平行四邊形區域。在此模型中，所有格點初始為空，並隨著時間從 0 漸進到 1逐漸被佔據。當某個佔據簇首次觸及 B(N) 的邊界時，即視為凍結。利用 [17] 中發展的技術，我們證明了一個比 Makowiec 在其猜想 3.1 中提出的結果更強的結論：當模型考慮在區域 B(N) 中時，原點在時間 1 屬於凍結簇的機率隨著 N → ∞趨近於零。 我們進一步引入並分析一個變體模型，稱為穿越凍結滲流模型。在此模型中，若某個佔據簇包含一條橫向（左到右）或縱向（上到下）貫穿 B(N) 的路徑，則該簇立即凍結。透過與邊界凍結滲流相同的方法，我們同樣證明：在此模型中，原點在時間 1 屬於凍結簇的機率也會隨著 N → ∞ 而趨近於零。

We study the boundary frozen percolation introduced by Makowiec in [7], which is defined on the finite subgraph B(N) of the triangular lattice. Here, B(N) denotes the parallelogram centered at the origin with side length 2N. In this model, all the sites are initially vacant and become occupied as time evolves from 0 to 1. An occupied cluster becomes frozen as soon as it touches the boundary of B(N). Using techniques developed in [17], we prove a result stronger than Conjecture 3.1 of [7]: the probability of the origin being frozen at time 1 goes to zero as N → ∞. We further introduce and analyze a variant model, called crossing frozen percolation, where occupied clusters freeze as soon as containing either a horizontal or vertical crossing of B(N). Applying the same methods, we show that in this model as well, the probability of the origin being frozen at time 1 tends to zero as N → ∞.