Phase Transitions and Scaling Limits in Lattice Models

相变是自然界中普遍存在的现象，从常见的液态水冻结成冰到超导等技术关键现象，自科学存在以来一直是数学和物理研究中的一个活跃话题，并继续构成重要的研究挑战。格子模型是为研究自然相变而构建的，是依赖于连续参数的图子集上的概率分布。这项研究项目将研究几个重要的晶格模型的相变，包括(1)约束渗流模型，它为图的每个顶点随机分配“0”或“1”，满足表示分子结构方面的约束；(2)自回避行走，它是图上的一条路径，最多访问每个顶点一次，作为长链聚合物的模型引入。这项工作将拓宽和深化这门学科的数学基础，在理论物理和化学方面具有潜在的重要应用。研究者的目标是发展关于某些晶格模型的相变的新理论。在渗流模型中施加约束通常会使模型失去随机单调性，在无约束的情况下，这对于研究无限团簇行为所描述的相变是至关重要的。该项目的一个目标是开发新的组合和概率技术来研究无限团簇的行为，而不使用随机单调性。由于非马尔科夫结构，列举自我回避行走通常是困难的。自回避游动的连通常数和指数是基本图的基本性质，但对于极少数图是可以识别的。这个项目的另一个目标是获得关于大类图的连通常数和指数的新信息，这可以通过分析图上的调和函数来实现。与二维中心极限定理类似的是某些晶格模型的高度函数在临界点，即相变发生时的极限形状行为。研究人员还计划使用复分析和代数几何的技术来研究极限形状。

Phase transitions, ranging from familiar occurrences such as liquid water freezing into ice to technologically critical phenomena such as transition to superconductivity, are ubiquitous throughout nature, have been an active topic in mathematical and physical research for as long as science has existed, and continue to pose important research challenges. Lattice models, constructed to study natural phase transitions, are probability distributions on subsets of graphs depending on continuous parameters. This research project will study the phase transition of several important lattice models, including (1) the constrained percolation model, which randomly assigns "0" or "1" to each vertex of a graph, satisfying constraints that represent aspects of molecular structure; and (2) the self-avoiding walk, which is a path on a graph visiting each vertex at most once, introduced as a model for long-chain polymers. The work will broaden and deepen the mathematical foundations of the subject, with potential important applications in theoretical physics and chemistry.The investigator aims to develop new theory concerning the phase transition of certain lattice models. Imposing constraints in the percolation model usually makes the model lose stochastic monotonicity, which, in the unconstrained case, is critical to study the phase transition described by the behavior of infinite clusters. One goal of the project is to develop new combinatorial and probabilistic techniques to study the behavior of infinite clusters without using stochastic monotonicity. Enumerating self-avoiding walks is typically difficult due to the non-Markovian structure. The connective constant and exponent of self-avoiding walks are fundamental properties of the underlying graph, yet can be identified for very few graphs. Another goal of this project is to obtain new information about connective constants and exponents for large classes of graphs, which may be achieved by analyzing harmonic functions on graphs. An analog to the central limit theorem in two dimensions is the limit shape behavior of height functions of certain lattice models at criticality, i.e. when the phase transition occurs. The investigator also plans to study the limit shape, using techniques from complex analysis and algebraic geometry.