The Toda Lattice and Universality for the Computation of the Eigenvalues of a Random Matrix

Written by leaders in the field, this text showcases some of the remarkable properties of the finite Toda lattice and applies this theory to establish universality for the associated Toda eigenvalue algorithm for random Hermitian matrices. The authors expand on a 2019 course at the Courant Institute to provide a comprehensive introduction to the area, including previously unpublished results. They begin with a brief overview of Hamiltonian mechanics and symplectic manifolds, then derive the action-angle variables for the Toda lattice on symmetric matrices. This text is one of the first to feature a new perspective on the Toda lattice that does not use the Hamiltonian structure to analyze its dynamics. Finally, portions of the above theory are combined with random matrix theory to establish universality for the runtime of the associated Toda algorithm for eigenvalue computation.

Percy Deift is Silver Professor of Mathematics at New York University. He is a Fellow of the American Mathematical Society, a member of the American Academy of Arts and Sciences, a member of the National Academy of Sciences and a member of the American Academy of Sciences and Letters. He is a winner of the George Pólya Prize, a Guggenheim Fellow and a winner of the Henri Poincare Prize.

Guillaume Dubach is Professeur Monge at École Polytechnique, Paris. He was awarded the 2019 Wilhelm Magnus Prize and the 2024 Shiing-Shen Chern Young Faculty Award.

Carlos Tomei is Professor in the Department of Mathematics at the Pontifical Catholic University of Rio de Janeiro (PUC-Rio). He is a member of the Brazilian Academy of Sciences and a SIAM Fellow.

Thomas Trogdon is Professor in the Department of Applied Mathematics at the University of Washington. He won the 2013 SIAM Richard C. DiPrima Prize, the 2017 SIAM Gábor Szegő Prize and earned an NSF CAREER award. He coauthored the book 'Riemann-Hilbert Problems, Their Numerical Solution, and the Computation of Nonlinear Special Functions' (2016) with Sheehan Olver.