A discrete-time Matsumoto-Yor theorem

• Charlie Herent (Université Paris-Saclay)

We study a random walk on the subgroup of lower triangular matrices of SL2, with i.i.d. increments. We prove that the process of the lower corner of the random walk satisfies a Rogers-Pitman criterion to be a Markov chain if and only if the increments are distributed according to a Generalized Inverse Gaussian (GIG) law on their diagonals. For this, we prove a new characterization of these laws. We prove a discrete-time version of the Dufresne identity. We show how to recover the Matsumoto-Yor theorem by taking the continuous limit of the random walk.

A probabilistic model for interpolation Macdonald polynomials

• Houcine Ben Dali (Harvard University)

Previous work of Ayyer, Martin, and Williams gave a probabilistic interpretation of the Macdonald polynomials at q=1 in terms of a Markov chain called the multispecies t-Push TASEP. Knop and Sahi introduced a family of inhomogeneous polynomials–defined by vanishing conditions–called Interpolation Macdonald polynomials, and from which classical Macdonald polynomials are recovered by taking the top homogeneous part. I will present a new Markov chain called the interpolation $t$-Push TASEP whose partition function corresponds to the interpolation Macdonald polynomial, evaluated at q=1. This generalizes the previous result of Ayyer, Martin, and Williams. This is based on a joint work with Lauren Williams.