On models of random permutations with cycle weights and spatial structure

关于具有循环权重和空间结构的随机排列模型

• 批准号: EP/G056390/1
• 负责人: Daniel Ueltschi
• 金额: $ 31.56万
• 依托单位: University of Warwick
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2010
• 资助国家: 英国
• 起止时间: 2010 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FG056390%2F1
• 关键词: models; random; permutations; cycle; weights

Random permutations have been a hot topic in probability theory and applied mathematics because of their connections to random matrices and other systems. The present study goes beyond the uniform permutations that have been considered so far. Of interest for us will be models where permutations receive a weight that depends on their cycle distribution. Another model of interest involves permutations of points in the space, with a weight that depends on the length of the permutation jumps. The main question deals with the possible occurrence of infinite cycles.These models find their origin in two unrelated areas. First, they describe the phenomenon of Bose-Einstein condensation, which is a phase transition that occurs for large systems of bosonic quantum particles. Second, they describe alleles in haploid populations, which are very much studied in mathematical biology.Results are expected to have consequences beyond mathematics. In particular, they will help the physicists to understand the effects of the particle interactions on the Bose-Einstein condensation. Indeed, models of spatial permutations are simpler, yet they describe quantum systems exactly. Thus they will allow for more detailed results.

随机排列是概率理论和应用数学的热门话题，因为它们与随机矩阵和其他系统的联系。本研究超出了迄今为止考虑的统一排列。对我们感兴趣的是置换量取决于其周期分布的重量的模型。感兴趣的另一个模型涉及空间中点的排列，重量取决于置换的长度跳跃。主要问题涉及无限周期的可能发生。这些模型在两个无关的地区找到了起源。首先，他们描述了Bose-Einstein凝结的现象，这是用于大型玻体量子颗粒系统的相变。其次，他们描述了单倍体种群中的等位基因，这些等位基因在数学生物学中进行了很多研究。预期将产生数学以外的后果。特别是，它们将帮助物理学家了解粒子相互作用对玻色网凝结的影响。实际上，空间排列的模型更简单，但它们准确地描述了量子系统。因此，它们将允许更详细的结果。

项目成果

Multispecies Virial Expansions

多物种病毒扩张

• DOI: 10.1007/s00220-014-2026-9
• 发表时间: 2014
• 期刊: Communications in Mathematical Physics
• 影响因子: 2.4
• 作者: Jansen S

Lattice permutations and Poisson-Dirichlet distribution of cycle lengths

晶格排列和周期长度的泊松-狄利克雷分布

• DOI: 10.48550/arxiv.1107.5215
• 发表时间: 2011
• 作者: Grosskinsky S

Virial Expansion Bounds Through Tree Partition Schemes

通过树分区方案进行病毒扩张

• DOI: 10.48550/arxiv.1501.00509
• 发表时间: 2015
• 作者: Ramawadh S

Cycle structure of random permutations with cycle weights

具有循环权重的随机排列的循环结构

• DOI: 10.48550/arxiv.1102.4796
• 发表时间: 2011
• 作者: Ercolani N

Spatial random permutations and Poisson-Dirichlet law of cycle lengths

空间随机排列和周期长度的泊松-狄利克雷定律

• DOI: 10.48550/arxiv.1007.2224
• 发表时间: 2010
• 作者: Betz V