Scattering of the Toda system and the Gaussian $\beta$-ensemble

The classical Toda flow is a well-known integrable Hamiltonian system that diagonalizes matrices. By keeping track of the distribution of entries and precise scattering asymptotics, one can exhibit matrix models for log-gases on the real line. These types of scattering asymptotics date back to fundamental work of Moser. More precisely, using the classical Toda flow acting on symmetric real tridiagonal matrices, we give a "symplectic" proof of the fact that the Dumitriu-Edelman tridiagonal model has a spectrum following the Gaussian $\beta$-ensemble.