Superconcentration and Related Topics
Superconcentration and Related Topics
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DOI：
10.1007/978-3-319-03886-5
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发表时间：
2014-01
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影响因子：
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通讯作者：
S. Chatterjee
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作者：
S. Chatterjee
理解随机对象的波动是概率论的主要目标之一。概率和分析有一个完整的子领域，称为测量集中，致力于理解随机对象的波动。在过去的四十年里，测量集中取得了巨大的进步。然而，有一大类问题，其中经典浓度的措施，使次优的范围内的顺序波动。在2008年和2009年，我在arXiv上发布了两个预印本，其中表明经典浓度的次优性，当它发生时，不仅仅是一个数学不足的问题。次优性实际上等同于，在被研究的随机对象的结构中，发生的一些非常有趣的事情。事实上，其结果可能足够有趣，以至于经典浓度的次优性值得拥有自己的名字;我称之为“超浓度”。这本专著是这两本预印本（不会单独出版）的结合，以及一些新材料和新见解。大多数结果与预印本相同，但呈现方式完全不同。特别是，我认为通过使用光谱方法，我实现了相当程度的简化和清晰。这在噪声敏感性文献中是相当标准的（这与本专题的主题密切相关），但这不是我在预印本中得出结果的方式。除了定理和证明，我还在文档中穿插了相当数量的专业数学家的开放问题和研究生的练习。
Understanding the fluctuations of random objects is one of the major goals of probability theory. There is a whole subfield of probability and analysis, called concentration of measure, devoted to understanding fluctuations of random objects. Measure concentration has seen tremendous progress in the last forty years. And yet, there is a large class of problems in which classical concentration of measure gives suboptimal bounds on the order of fluctuations. In 2008 and 2009, I posted two preprints on arXiv where it was shown that the suboptimality of classical concentration, when it occurs, is not simply a question of mathematical inadequacy. The suboptimality is in fact equivalent to a number of very interesting things going on in the structure of the random object under investigation. Indeed, the consequences are possibly interesting enough for the suboptimality of classical concentration to deserve a name of its own; I call it ‘superconcentration’.This monograph is a combination of these two preprints (which will not be published individually), together with some new material and new insights. The majority of the results are the same as in the preprints, but the presentation is radically different. In particular, I think I achieved a substantial degree of simplification and clarity through the use of the spectral approach. This is quite standard in the noisesensitivity literature (which is intimately connected with the topic of this monograph), but it is not the way I derived the results in the preprints. In addition to the theorems and proofs, I have interspersed the document with a sizable number of open problems for professional mathematicians and exercises for graduate students.