相互作用粒子系统中的动态现象：相变与平衡态
Dynamic Phenomena in Interacting Particle Systems: Phase Transitions and Equilibrium
AI总结：
本论文通过三项研究探讨随机模型的临界现象与平衡态，分析了高密度一维环上 Activated Random Walk 的慢相存在性、修正接触过程的不变测度及弱收敛性，以及次临界分支过程的准平稳分布存在唯一性。
摘要：
本论文通过三项相互关联的研究，探讨各种随机模型中的临界现象与平衡态。第一章，我们分析高密度区域中一维环上的 Activated Random Walk 模型。我们引入一种逐步构建环境的 toppling 程序，展示了长时间内的持续活动性。该方法为任意大 sleep rate 下慢相的存在性提供了简洁且自洽的证明。第二章聚焦于具有可变感染率的修正一维接触过程。具体而言，感染在感染区域边界处以 $λ_e$ 速率传播，在其他区域以 $λ_i$ 速率传播。我们证明了当 $λ_i=λ_c$、$λ_e=λ_c+\varepsilon$ 时该过程存在不变测度，其中 $λ_c$ 表示标准接触过程的临界参数。此外，我们证明从右边缘观察时，该过程弱收敛于此不变测度。我们还表明在相空间的吸引区域内，感染沿临界曲线几乎必然消亡。最后一章，我们探讨连续时间下两个次临界种群过程的准平稳分布（QSDs）：分支随机游走和带系谱的分支过程。我们利用其动力学的空间特性，证明了这些过程 QSDs 的存在性与唯一性。
英文摘要：
This thesis investigates critical phenomena and equilibrium states in various stochastic models through three interconnected studies. In the first chapter, we analyze the Activated Random Walk model on a one-dimensional ring in the high-density regime. We introduce a toppling procedure that incrementally constructs an environment demonstrating the sustained activity over extended periods. This approach provides a concise and self-contained proof of the existence of a slow phase for arbitrarily large sleep rates. The second chapter focuses on a modified unidimensional contact process with varying infection rates. Specifically, infection spreads at rate $λ_e$ at the boundaries of the infected region and at rate $λ_i$ elsewhere. We establish the existence of an invariant measure for this process when $λ_i=λ_c$, $λ_e=λ_c+\varepsilon$ where $λ_c$ denotes the critical parameter for the standard contact process. Furthermore, we demonstrate that the process, when observed from the right edge, converges weakly to this invariant measure. We also show that infection dies almost surely along the critical curve within the attractive region of the phase space. In the final chapter, we explore quasi-stationary distributions (QSDs) for two subcritical population processes in continuous time: branching random walks and branching processes with genealogy. We prove the existence and uniqueness of QSDs for these processes by leveraging spatial aspects of their dynamics.
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