Titlebook: Superconcentration and Related Topics; Sourav Chatterjee Book 2014 Springer International Publishing Switzerland 2014 60E15, 60K35, 60G15,

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Sourav Chatterjees of interactions between people from different cultures have contributed to the appreciation of diverse viewpoints. While the sudden emergence and proliferation of the internet has altered the landscape of relationships and social interactions, there has not been sufficient time to revisit and stud

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Sourav Chatterjeeisches Debüt. Sie thematisiert einen kurzen Besuch in der Heimatstadt des Erzählers, die er vier Jahre vorher verlassen hat. Es ist der Versuch einer Heimkehr, der Aussöhnung des Protagonisten mit seiner Geschichte als Voraussetzung für eine selbstbewußte Standortbestimmung des Ichs in der Gegenwart

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978-3-319-35228-2Springer International Publishing Switzerland 2014

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Markov Semigroups,This chapter contains a quick review of some basic facts and definitions related to Markov semigroups, such as the heat equation, the Ornstein-Uhlenbeck semigroup, Poincaré inequalities, Fourier expansions in the orthogonal basis of eigenfunctions of a semigroup, and some basic applications of these tools.

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Superconcentration and Chaos,This chapter contains the formal definitions of superconcentration and chaos, and a theorem proving the equivalence of superconcentration and chaos. This is one of the key results of this monograph. Applications of the equivalence theorem to prove chaos in polymers, spin glasses and eigenvectors of random matrices are worked out.

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Multiple Valleys,This chapter gives the formal definition of multiple valleys, and the proof that chaos implies multiple valleys. This is another key result of this monograph. This result is then used to prove multiple valleys in polymer models, spin glasses and the Kauffman-Levin fitness model from evolutionary biology.

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The Spectral Method for Proving Superconcentration,This chapter introduces a new method for proving superconcentration, based on an improved Poincaré inequality. The author calls this the “spectral method”. The spectral method is used to give a short proof of superconcentration in the Sherrington-Kirkpatrick model, substantially simplifying the original proof given by the author some years ago.

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Extremal Fields,This chapter introduces the notion of extremal Gaussian fields and proves superconcentration in extremal fields. The method is then used to prove superconcentration in certain kinds of spin glass models and in the discrete Gaussian free field.