Independence Complexes of Cylinders Constructed from Square and Hexagonal Grid Graphs
Independence Complexes of Cylinders Constructed from Square and Hexagonal Grid Graphs
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由正方形和六边形网格图构造的圆柱体独立复合体
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发表时间：
2008
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通讯作者：
Johan Thapper
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作者：
Johan Thapper
组合学和统计力学之间的相互作用在这两个领域提供了许多富有成效的见解。一个令人信服的例子是Kuperberg对交替符号矩阵猜想的解决方案及其以下概括。在这篇论文中，我们研究了近年来受到关注的两个统计力学模型。第一种是完全填充回路模型。Razumov和Stroganov在2001年提出的一个猜想为O（1）循环模型及其与完全填充循环模型的改进之间的联系的大规模持续研究开辟了领域。我们应用组合双射最初发现德吉尔的一个老猜想提出的Propp。第二个模型是硬粒子模型。最近Fendley等人的发现和Jonsson的结果表明，具有圆柱边界条件的硬正方形模型具有一些美丽的组合性质。我们应用拓扑和纯组合的方法，相关的独立复杂的尝试，并获得更好的理解这个模型。
Interactions between combinatorics and statistical mechanics have provided many fruitful insights in both fields. A compelling example is Kuperberg’s solution to the alternating sign matrix conjecture, and its following generalisations. In this thesis we investigate two models from statistical mechanics which have received attention in recent years. The first is the fully packed loop model. A conjecture from 2001 by Razumov and Stroganov opened the field for a large ongoing investigation of the O(1) loop model and its connections to a refinement of the fully packed loop model. We apply a combinatorial bijection originally found by de Gier to an older conjecture made by Propp. The second model is the hard particle model. Recent discoveries by Fendley et al. and results by Jonsson suggests that the hard square model with cylindrical boundary conditions possess some beautiful combinatorial properties. We apply both topological and purely combinatorial methods to related independence complexes to try and gain a better understanding of this model.