Dynamics of the McKean-Vlasov Equation

Abstract

This note studies the deterministic flow of measures which is the limiting case as $n \rightarrow \infty$ of Dyson's model of the motion of the eigenvalues of random symmetric $n \times n$ matrices. Though this flow is nonlinear, highly singular and apparently of Wiener-Hopf type, it may be solved explicitly without recourse to Wiener-Hopf theory. The solution greatly clarifies the role of the famous Wigner semicircle law.