Probability seminar: Daniel Perales

Daniel Perales
University of Notre Dame

Room: M3 3127

Finite Free Probability

The finite free convolutions are binary operations on polynomials that behave well with respect to the roots and can be understood as a discrete analogue of free probability. On the other hand, these operations can be realized as expected characteristic polynomials of adding (or multiplying) two randomly rotated matrices. We will survey some interesting applications: 

• Constructing hypergeometric polynomials with real roots and applications to multiple orthogonality (arXiv:2309.10970, arXiv:2404.11479).
• Understanding the behavior of roots under repeated differentiation (arxiv:2108.08489).
• Asymptotic behavior of the ratio of consecutive coefficients and Voiculescu's S-transform (arXiv:2408.09337).
• The finite free commutator and even polynomials (arXiv:2502.00254).

Finally I will also mention ongoing work and interesting future directions.

Based on joint works with many authors: Octavio Arizmendi, Jacob Campbell, Katsunori Fujie, Jorge Garza-Vargas, Andrei Martinez-Finkelshtein, Rafael Morales and Yuki Ueda.