Sylvia Serfaty 12 Microscopic description of Log and Coulomb gases part 12

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Sylvia Serfaty, New York University. Part 3. Lecture notes available at https://pcmi.ias.edu/sites/pcmi.ias.e... • As observed by Dyson and Wigner, instances of classical random matrix ensembles (such as the Gaussian Unitary Ensemble, Gaussian Orthogonal Coulomb Gases, Ginibre Ensemble) can also be viewed as systems of particles in the plane or on the real line with logarithmic or Coulomb interactions, at particular temperatures, which are also called beta-ensembles. These have been in recent years intensely studied for their own sake. We will examine general Coulomb and Log gases (including in higher dimension than 2), taking a point of view based on the detailed expansion of the interaction energy. This allows us to describe the macroscopic and microscopic behavior of the systems. In particular we will show a Large Deviations Principle for the empirical field and a Central Limit Theorem for fluctuations down to the mesoscopic scales. This allows us to observe the effect of the temperature as it gets very large or very small, and to connect with crystallization questions, such as the occurence of the triangular Abrikosov lattice. The main results are joint with Thomas Leble and also based on previous works with Etienne Sandier, Nicolas Rougerie and Mircea Petrache. • Presented at the 27th Annual PCMI Summer Session, Random Matrices, held June 25 – July 15, 2017. The residential, three-week Summer Session is the flagship activity of the IAS/Park City Mathematics Institute (PCMI). • About PCMI • The Institute for Advanced Study / IAS / Park City Mathematics Institute (PCMI) is designed for mathematics educators at the secondary and post-secondary level, as well as mathematics researchers and students at the post-secondary level. These groups find at PCMI an intensive mathematical experience geared to their individual needs. Moreover, the interaction among groups with different backgrounds and professional needs increases each participant’s appreciation of the mathematical community as a whole as well as the work of participants in different areas. • For more information, visit https://pcmi.ias.edu

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