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020 _a9783319038865
_9978-3-319-03886-5
024 7 _a10.1007/978-3-319-03886-5
_2doi
050 4 _aQA273.A1-274.9
050 4 _aQA274-274.9
072 7 _aPBT
_2bicssc
072 7 _aPBWL
_2bicssc
072 7 _aMAT029000
_2bisacsh
082 0 4 _a519.2
_223
100 1 _aChatterjee, Sourav.
_eauthor.
245 1 0 _aSuperconcentration and Related Topics
_h[electronic resource] /
_cby Sourav Chatterjee.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2014.
300 _aIX, 156 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aSpringer Monographs in Mathematics,
_x1439-7382
505 0 _aPreface -- 1.Introduction -- 2.Markov semigroups -- 3.Super concentration and chaos -- 4.Multiple valleys -- 5.Talagrand’s method for proving super concentration -- 6.The spectral method for proving super concentration -- 7.Independent flips -- 8.Extremal fields -- 9.Further applications of hypercontractivity -- 10.The interpolation method for proving chaos -- 11.Variance lower bounds -- 12.Dimensions of level sets -- Appendix A. Gaussian random variables -- Appendix B. Hypercontractivity -- Bibliography -- Indices.
520 _aA certain curious feature of random objects, introduced by the author as “super concentration,” and two related topics, “chaos” and “multiple valleys,” are highlighted in this book. Although super concentration has established itself as a recognized feature in a number of areas of probability theory in the last twenty years (under a variety of names), the author was the first to discover and explore its connections with chaos and multiple valleys. He achieves a substantial degree of simplification and clarity in the presentation of these findings by using the spectral approach. Understanding the fluctuations of random objects is one of the major goals of probability theory and a whole subfield of probability and analysis, called concentration of measure, is devoted to understanding these fluctuations. This subfield offers a range of tools for computing upper bounds on the orders of fluctuations of very complicated random variables. Usually, concentration of measure is useful when more direct problem-specific approaches fail; as a result, it has massively gained acceptance over the last forty years. And yet, there is a large class of problems in which classical concentration of measure produces suboptimal bounds on the order of fluctuations. Here lies the substantial contribution of this book, which developed from a set of six lectures the author first held at the Cornell Probability Summer School in July 2012. The book is interspersed with a sizable number of open problems for professional mathematicians as well as exercises for graduate students working in the fields of probability theory and mathematical physics. The material is accessible to anyone who has attended a graduate course in probability.
650 0 _aMathematics.
650 0 _aDistribution (Probability theory).
650 1 4 _aMathematics.
650 2 4 _aProbability Theory and Stochastic Processes.
650 2 4 _aMathematical Physics.
650 2 4 _aStatistical Physics, Dynamical Systems and Complexity.
710 2 _aSpringerLink (Online service)
773 0 _tSpringer eBooks
776 0 8 _iPrinted edition:
_z9783319038858
830 0 _aSpringer Monographs in Mathematics,
_x1439-7382
856 4 0 _uhttp://dx.doi.org/10.1007/978-3-319-03886-5
912 _aZDB-2-SMA
999 _c93040
_d93040