2021/11/10 by Tristan Gautié, Gautié, Tristan · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #Quantum Physics (quant-ph) #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2111.05737
openalex publication_date 2021/11/10 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
This statistical physics thesis focuses on the study of three kinds of\nsystems which display repulsive interactions: eigenvalues of random matrices,\nnon-crossing random walks and trapped fermions. These systems share many links,\nwhich can be exhibited not only at the level of their static version, but also\nat the level of their dynamical version. We present a combined analysis of\nthese systems, employing tools of random matrix theory and stochastic calculus\nas well as tools of quantum mechanics, in order to solve some original\nproblems. Further from the detailed presentation of the field and the report of\nthe results obtained during the PhD, the different themes exposed in the\nchapters of the thesis allow for perspectives on related issues.\n As such, the first chapter is an introduction to random matrix theory; we\ndetail its historical evolution and numerous applications, and present its\nessential concepts, constructions and results. The second chapter discusses\nnon-crossing random walks; we describe the deep links they share with random\nmatrix eigenvalue processes and showcase the results obtained in the scope of\nboundary problems. In the third chapter, which focuses on stochastic matrix\nprocesses, we introduce in particular a process inspired from the Kesten random\nrecursion, and highlight the new link it allows to draw between the\ninverse-Wishart ensemble and fermions trapped in the Morse potential. Lastly,\nthe fourth chapter, centred on the particular case of bridge processes, allows\nfor a joint treatment of scalar and matrix models; therein, we develop a\ngeneralization of the Ferrari-Spohn problem for non-crossing scalar bridges\nand, as an opening, we exhibit the connections of matrix bridges with other\naspects of random matrices.\n