Marta Dell’Atti – Wednesday 10 June 2026

Speaker: Marta Dell’Atti (University of Warsaw)

Title: Even Pfaff lattice and skew-orthogonal polynomials

Abstract:

Motivated by the theory of orthogonal random matrix ensembles, we investigate the integrable semi-discrete hierarchy of the Pfaff lattice. The focus on the even times of the hierarchy allows us to introduce the Even Pfaff lattice, similarly as the Volterra lattice can be deduced from the Toda lattice, the latter associated with the unitary random matrix ensemble. Via a map connecting Toda and Pfaff we explicitly determine the elements populating the Even Pfaff lattice at the initial time (with the Gauss weight). In [1] the initial datum provides the natural approach to construct the continuum limit, while also suggesting an interesting natural reduction of the system. The map exists at the level of the eigenfunctions of the semi-infinite Lax operator: orthogonal polynomials for Volterra and skew-orthogonal polynomials for Even Pfaff. In [2] we derive some closed relations for the skew-orthogonal polynomials for the Gauss weight, and determine the map and the modified three-term relations for the skew-orthogonal polynomials associated with the quartic Freud weight.

The talk is based on the work in collaboration with C. Benassi and A. Moro.

References

1. C. Benassi, M. Dell’Atti, A. Moro: Random matrix ensemble and integrable differential identities (2026), preprint on arXiv:2504.11296 [math-ph].
2. C. Benassi, M. Dell’Atti: Skew-orthogonal polynomials for a quartic Freud weight: two classes of quasi-orthogonal polynomials (2026), preprint on arXiv:2604.22616 [math.CA].