  This statistical physics thesis focuses on the study of three kinds of
systems which display repulsive interactions: eigenvalues of random matrices,
non-crossing random walks and trapped fermions. These systems share many links,
which can be exhibited not only at the level of their static version, but also
at the level of their dynamical version. We present a combined analysis of
these systems, employing tools of random matrix theory and stochastic calculus
as well as tools of quantum mechanics, in order to solve some original
problems. Further from the detailed presentation of the field and the report of
the results obtained during the PhD, the different themes exposed in the
chapters of the thesis allow for perspectives on related issues.
  As such, the first chapter is an introduction to random matrix theory; we
detail its historical evolution and numerous applications, and present its
essential concepts, constructions and results. The second chapter discusses
non-crossing random walks; we describe the deep links they share with random
matrix eigenvalue processes and showcase the results obtained in the scope of
boundary problems. In the third chapter, which focuses on stochastic matrix
processes, we introduce in particular a process inspired from the Kesten random
recursion, and highlight the new link it allows to draw between the
inverse-Wishart ensemble and fermions trapped in the Morse potential. Lastly,
the fourth chapter, centred on the particular case of bridge processes, allows
for a joint treatment of scalar and matrix models; therein, we develop a
generalization of the Ferrari-Spohn problem for non-crossing scalar bridges
and, as an opening, we exhibit the connections of matrix bridges with other
aspects of random matrices.
