Théorie des matrices aléatoires en physique statistique : théorie quantique de la diffusion et systèmes désordonnés

Abstract : Random matrix theory has applications in various fields: mathematics, physics, finance, ... In physics, the concept of random matrices has been used to study the electronic transport in mesoscopic structures, disordered systems, quantum entanglement, interface models in statistical physics, cold atoms, ... In this thesis, we study coherent AC transport in a quantum dot, properties of fluctuating 1D interfaces on a substrate and topological properties of multichannel quantum wires. The first part gives a general introduction to random matrices and to the main method used in this thesis: the Coulomb gas. This technique allows to study the distribution of observables which take the form of linear statistics of the eigenvalues. These linear statistics represent many relevant physical observables, in different contexts. This method is then applied to study concrete examples in coherent transport and fluctuating interfaces in statistical physics. The second part focuses on a model of disordered wires: the multichannel Dirac equation with a random mass. We present an extension of the powerful methods used for one dimensional system to this quasi-1D situation, and establish a link with a random matrix model. From this result, we extract the density of states and the localization properties of the system. Finally, we show that this system exhibits a series of topological phase transitions (change of a quantum number of topological nature, without changing the symmetries), driven by the disorder.
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Aurélien Grabsch. Théorie des matrices aléatoires en physique statistique : théorie quantique de la diffusion et systèmes désordonnés. Quantum Physics [quant-ph]. Université Paris-Saclay, 2018. English. ⟨NNT : 2018SACLS142⟩. ⟨tel-01849097⟩

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