Mathematics Colloquium

The Toda lattice prescribes the evolution of N particles interacting under certain Hamiltonian dynamics; it is an archetypal example of a completely integrable system. A question of interest is to understand how the model behaves, under random (or typical) initial data, when the number N of particles becomes large. In this talk describe several results explaining such asymptotics. The proofs proceed by finding a way to interpret the Toda lattice (under certain random initial data) as a dense collection of solitons, and providing a framework to study how these solitons asymptotically evolve in time. In this analysis, Lyapunov exponents (arising from products of random matrices) play an important role.