Probability on graphs: random processes on graphs and lattices, by Geoffrey Grim-
Probability on graphs: random processes on graphs and lattices, by Geoffrey Grim-
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图上的概率：图和格上的随机过程，作者：Geoffrey Grim-
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发表时间：
2013
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通讯作者：
Geoffrey Grim
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作者：
Geoffrey Grim
被称为相互作用粒子系统（IPS）的数学领域的发展在很大程度上是由Liggett 1985年的专著[12]催化的。30年后，这个IPS领域（连同密切相关的渗流场）仍然是数学概率和统计物理学定理证明的活跃领域，其MSC主要分类60 K35每年包含约170篇论文。简而言之，IPS研究图上的模型：每个顶点（或站点）都处于某种状态;顶点与相邻顶点以随机间隔相互作用，并根据某种规则更新它们的状态。例如，在接触过程中，一个流行病的自然玩具模型，一个顶点要么被感染，要么是健康的：健康的站点以λ乘以被感染的邻居数量的速率被感染，而被感染的站点以恒定的速率1（速率意味着单位时间的概率）变得健康。Liggett的书对过去15年中几个模型的工作进行了出色的描述，其中一个主要的原始动机是严格研究相变的统计物理概念，形式化为定义在无限d维晶格上的过程的行为的定性变化，因为参数通过临界值。例如，在接触过程中，如果λ小于临界值，则流行病将消亡，但如果λ较大，则有机会永远生存。IPS不仅在统计物理学和数学概率中仍然是一个活跃的领域，而且类似的模型已经在其他数量学科中使用。仅举两个最近的研究生教材，[14]涉及与计算复杂性（可满足性问题）和信息论（纠错码）相关的模型;[7]显示了渗流模型的连续模拟如何成为无线网络通信数学研究的基础。社交网络提供了一个更熟悉的环境。无论你认为社交网络做什么，它都涉及到一些信息交换的概念，玩具数学模型不可避免地与IPS的模型有些相似（例如，你的观点如何受到朋友观点的影响，选民模型有很多变体）。参见[3]对此类文献的广泛概述，以及[1]对IPS精神的新颖玩具模型。在所有这些更广泛的学科中，适当的基础“几何”（由图描述）是依赖于上下文的，而不是经典的物理模型中使用的三维晶格，如铁磁性现象。写一个后续的书类似于利格特的介绍性和全面似乎相当不切实际，不仅因为规模的领域，但也因为它需要复制许多利格特的材料。因此，后续的书籍文献包括一小部分课堂笔记和专门的专著，如[4，6，8，9，13]。不幸的是，这使得一个新来者很难知道如何在这个领域开始，如何开始学习今天研究前沿背后的技术、模型和结果。这本书的评论是
Growth of the field of mathematics called Interacting Particle Systems (IPS) was, to a great extent, catalyzed by Liggett’s 1985 monograph [12]. Thirty years later this IPS field (together with the closely related percolation field) remains an active area within mathematical probability and the theorem-proof end of statistical physics, with its MSC primary classification 60K35 containing about 170 papers a year. In brief, IPS studies models over a graph: each vertex (or site) is in some state; vertices interact with neighboring vertices at random intervals and update their states according to some rule. For instance in the contact process, a natural toy model for epidemics, a vertex is either infected or healthy: healthy sites become infected at rate λ times the number of infected neighbors, and infected sites become healthy at constant rate one (rate means probability per unit time). Liggett’s book gave a masterful account of work on several models over the previous 15 years, for which a major original motivation had been rigorous study of the statistical physics notion of phase transitions, formalized as qualitative changes in behavior of a process defined on the infinite d-dimensional lattice as parameters pass through a critical value. For instance in the contact process, the epidemic will die out if λ is smaller than the critical value, but has a chance to survive forever if λ is larger. Not only has IPS remained an active field within statistical physics and mathematical probability, but similar models have been used in a surprising range of other quantitative disciplines. To cite just two recent graduate texts, [14] treats models relevant to topics within computational complexity (satisfiability problems) and information theory (error-correcting codes); and [7] shows how continuum analogs of percolation models are fundamental to mathematical study of wireless network communication. A more familiar context is provided by social networks. Whatever you think a social network does, it involves some notion of exchange of information, and toy mathematical models are inevitably somewhat similar to the models from IPS (for instance there are many variants of the voter model for how your opinions are affected by your friends’ opinions). See [3] for a wide-ranging overview of such literature, and [1] for novel toy models in the spirit of IPS. In all these broader disciplines the appropriate underlying “geometry” (described by the graph) is context-dependent, not the 3-dimensional lattice used classically in physics models for phenomena such as ferromagnetism. Writing a subsequent book which resembled Liggett’s in being both introductory and comprehensive seems quite impractical, not only because of the size of the field but also because it would need to duplicate much of Liggett’s material. So the subsequent book literature consists of a small set of lecture notes and specialized monographs, such as [4, 6, 8, 9, 13]. This makes it unfortunately difficult for a newcomer to know how to get started in the field—how to start learning the techniques, models and results behind today’s research frontier. The book under review serves