Microscopic aspects of logarithmic interaction systems

The eigenvalues of certain random matrices behave miraculously like particles in a one-dimensional gas with logarithmic repulsion. The same system, but seen in two dimensions, is called a "one-component plasma" in the physics literature and is studied as a toy model of statistical mechanics, of particular interest for the difficulties posed by the singular, long-range nature of the interaction. In this lecture, we will present some methods for the mathematical study of these systems at the microscopic scale, seen as probabilistic and analytic objects. These include both classical tools (large deviation theory, variational calculus, measurement transport, gamma-convergence, empirical fields) and ad hoc constructions (squaring and regularization, explicit constructions in infinite volume, role of the electric field). This study is part of the program initiated by Étienne Sandier and Sylvia Serfaty.