定義於頂點之模型中的單調性、關聯不等式與相變現象

Abstract

本論文探討定義於頂點之模型的單調性性質、關聯不等式以及相變現象。我們于這些模型上建立了與隨機叢集模型（random-cluster model）相對應的單調性結果，並透過耦合的芒硝動力學（Glauber Dynamics）的方式驗證鐵磁性易辛模型（Ising model）確實滿足此性質。此外，我們亦為玻茨模型（Potts model）證明一個調整後的結果，既我們利用連續時間馬可夫鏈的耦合方法推導出一個與FKG 不等式類似的變形版本。 本論文亦將原先應用於鍵滲流模型（bond percolation）的決策樹（decision-tree）架構改良後應用於點滲流模型（site percolation）。藉由調整后對應的取樣映射，我們得到一個與OSSS 不等式類似的上界。該不等式在點滲流模型中的其中一個銳利相變中扮演關鍵角色。將其應用於連通事件，可以證明在無限頂點可遞圖上的無限開叢集出現的相變行為。

This thesis investigates monotonicity properties, correlation inequalities, and phase transitions in various models on vertices. We establish monotonicity properties analogous to those known for the random-cluster model. We verify this property for the ferromagnetic Ising model via a coupled Glauber dynamics construction and prove an adapted result for the Potts model, where a continuous-time Markov chain coupling yields a correlation inequality that is similar to the FKG inequality. We also adapt the decision-tree framework that has been used for the bond percolation model to the site percolation model. By adapting the associated sampling map, we obtain a bound analogous to the OSSS inequality. This inequality plays a key role in demonstrating a sharp phase transition in the model. In particular, applying it to connectivity events indicate a phase transition in the existence of an infinite open cluster on infinite vertex-transitive graphs.