Phase transitions in random graphs and random graph processes

在过去的几十年里，组合结构得到了广泛的研究，并成为当代数学的中心主题之一。特别是对随机图的研究汇集了不同的领域，如离散数学、概率论、理论计算机科学和统计物理。本课题的目标是研究随机图和随机图过程中具有程度分布、禁止子结构、属等约束的相变。在统计物理、数学和理论计算机科学的许多基本问题中都观察到相变现象，包括波茨模型、图形着色和可满足性问题。在大量不同的随机图模型中观察到的相变是指这样一种现象，即存在一个边缘密度临界值，在临界值周围添加少量边缘会导致最大分量的大小发生巨大变化。我们的目标是进一步发展和应用新的分析方法，结合计数和概率方法，如奇点分析，微分方程方法，以研究随机图和随机图过程中的相变。

Combinatorial structures have been extensively studied during the last few decades and have become one of the central themes of contemporary mathematics. The study of random graphs in particular has brought together different fields such as discrete mathematics, probability theory, theoretical computer science and statistical physics. The objectives of this project are to study the phase transitions in random graphs and random graph processes with constraints such as degree distribution, forbidden substructures, genus. The phase transition is a phenomenon observed in many fundamental problems from statistical physics, mathematics and theoretical computer science, including Potts models, graph colourings and satisfiability problem. The phase transition observed in the plethora of different random graph models refers to a phenomenon that there is a critical value of edge density such that adding a small number of edges around the critical value results in a dramatic change in the size of the largest components. It is our aim to further develop and apply new analytic approaches combined with counting and probabilistic methods, e.g. singularity analysis, differential equations method, to the study of the phase transitions in random graphs and random graph processes.