A Lax difference operator realization of the Pfaff lattice: generalization and applications
A talk in the Seminar Zufallsmatrizen series by
Sylvain Carpentier
| Abstract: | The Pfaff lattice provides an integrable description of deformations of skew-orthogonal polynomials and plays an important role in random matrix theory. In this talk, I will present a realization of this hierarchy using $ 2 \times 2 $ matrix pseudodifference operators, based on joint work with Marta Dell’Atti. Dressing the full matrix Laurent algebra leads to a broader framework relating the Pfaff lattice and Pfaff–Toda, with Toda hierarchies arising through reductions. I will also discuss the multicomponent extension. I will then explain how this formulation can be used to construct a quadratic Poisson bracket for the Pfaff lattice. Finally, I will describe ongoing work on rational reductions, which yield explicit scalar Lax operators and integrable lattices with finitely many fields. I will discuss their relation to moment matrices and polynomial recurrences, taking the Bures ensemble as a starting example, and outline how finite-rank moment constraints suggest further connections with random matrix models. Reference: S. Carpentier and M. Dell’Atti, Matrix Dressing Beyond Pfaff–Toda, arXiv:2609.12970 (2026). Within the CRC this talk is associated to the project(s): C6 |