发展高效蒙特卡洛算法研究逾渗类临界系统

There are a lot of phase transition phenomena in nature, ordinary percolation (OP) and its variants (i.e., directed percolation, DP) represent some of the basic classes. The study of percolation has important scientific significance and application value. Although the definitions of percolation models are very simple, very few of them have analytical solutions. Numerical methods become an important research tool. In the previous work, by means of Monte Carlo simulation, we determined the critical points of a number of OP and DP models, and analyzed the OP structures at criticality on square lattice. This project aims to extend some preliminary work, and by developing efficient algorithms to study the susceptible-infected-removed (SIR) model with long-range infection and the explosive percolation (EP) model. The content includes: 1) the fractal dimension of the leaf-free clusters generated by worm-type algorithm and the DP probability distributions; 2) phase diagram of the three-dimensional SIR model with long-range infection; 3) identification of the order of phase transition for the EP model on simple-cubic lattice which satisfies the minimum product rule.

自然界中存在着大量的相变现象，普通逾渗(ordinary percolation, OP)及其变种(如,directed percolation, DP)代表了其中一些基本类。对逾渗的研究具有重要的科学意义和应用价值。尽管逾渗模型的定义非常简单，但极少有解析解。数值方法成为重要的研究工具。前期工作中，利用蒙特卡洛模拟，我们已对诸多OP、DP模型的临界点进行了精确确定，并对正方格上OP的临界集团结构进行了探讨。本项目旨在延续部分前期工作，并发展高效算法研究长程susceptible-infected-removed (SIR)模型和explosive percolation (EP)模型。内容包括：1. 研究虫子型算法生成的leaf-free临界集团的分形维以及DP的概率分布；2.研究三维长程SIR模型的相图；3. 对简立方格上遵循最小乘积规则的EP模型的相变类型进行鉴别。