Minicourse IV: Asymptotic Spectral Analysis of Growing Graphs —— a quantum probability point of view
Minicourse IV: Asymptotic Spectral Analysis of Growing Graphs —— a quantum probability point of view
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发表时间：
2019
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通讯作者：
N. Obata
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作者：
N. Obata
这些讲座从量子概率的角度提供了一些关于增长（或大）图的渐近谱分析的主题。所谓生长图，我们指的是顶点数趋于无穷大的一系列图。实际上，我们主要研究图的邻接矩阵的谱，并研究它们在图增长时的渐近性。因此，我们的主题自然站在概率，分析和组合学的十字路口，也符合Vershik的渐近组合学思想[34，35]。此外，我们还想到了对复杂网络数学理论的贡献，例如，[10，11，23]。量子（或非对易）概率论，可以追溯到冯·诺依曼的著名工作[37]，其最初的目的是解决量子力学的统计问题，最近已被确立为非对易分析的新数学范式。随着经典概率论中的概念和技术如大数定律、中心极限定理、大偏差原理等被成功地应用于渐近问题，量子概率自然被期望在探索非对易系统的统计性质中发挥重要作用。在这些讲座中，我们看到量子概率的思想和方法是如何应用到图的谱分析，特别是增长图。
These lectures provide some topics in asymptotic spectral analysis of growing (or large) graphs from a quantum probability point of view. By growing graphs we mean just a sequence of graphs where the number of vertices tends to the infinity. In fact, we focus on spectra of the adjacency matrices of graphs and study their asymptotics when the graph grows. Thus our theme naturally stands at the crossroads of probability, analysis and combinatorics, and also in line with Vershik’s idea of asymptotic combinatorics [34, 35]. Moreover, we have in mind a contribution to the mathematical theory for complex networks, see e.g., [10, 11, 23]. The quantum (or non-commutative) probability theory, tracing back to the famous work by von Neumann [37], where the original aim was to solve statistical questions of quantum mechanics, has been recently established as a new mathematical paradigm for non-commutative analysis. As the concepts and techniques in classical probability theory such as the law of large numbers (LLN), central limit theorem (CLT), large deviation principle (LDP) and so forth have been successfully applied to asymptotic problems, quantum probability is naturally expected to play an essential role in exploring statistical properties of non-commutative systems. During these lectures we see how quantum probabilistic ideas and methods are applied to spectral analysis of graphs, in particular, of growing graphs.