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Résumé |
In this talk, I will consider the classical problem of linear statistics in random matrix theory. This amounts to study the distribution of the sum of a certain function of the matrix eigenvalues. Varying this function, this problem can describe fluctuations of conductance, shot noise, Renyi entropy, center of mass of interfaces, particle number…
This problem has been extensively studied for the bulk of the eigenvalues (macroscopic linear statistics) where interesting phase transitions have been unveiled but not so much at the edge of the spectrum (microscopic linear statistics) on which I will focus.
In particular, I will introduce four methods to solve this problem, show their equivalence and I will discuss the physical applications of these results (large deviations of the solution of the Kardar-Parisi-Zhang equation, existence of phase transitions with continuously varying exponent and possible experimental realization of this setup with non-intersecting Brownian interfaces).
Reference : Alexandre Krajenbrink & Pierre Le Doussal, Linear statistics and pushed Coulomb gas at the edge of beta random matrices: four paths to large deviations, preprint arXiv:1811.00509
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