Bangalore Probability Seminar : A multispecies totally asymmetric zero range process and Macdonald polynomials
Arvind Ayyer
Abstract
Macdonald polynomials are a remarkable family of symmetric
functions that are known to have connections to combinatorics, algebraic
geometry and representation theory. Due to work of Corteel, Mandelshtam
and Williams, it is known that they are related to the asymmetric simple
exclusion process (ASEP) on a ring.
The modified Macdonald polynomials are obtained from the Macdonald
polynomials using an operation called plethysm. It is natural to ask
whether the modified Macdonald polynomials are related to some other
particle system. In this talk, we answer this question in the
affirmative via a multispecies totally asymmetric zero-range process
(TAZRP). We also present a Markov process on tableaux that projects to
the TAZRP and derive formulas for stationary probabilities and certain
correlations. We also prove a remarkable symmetry property for local
correlations.
This is joint work with Olya Mandelshtam and James Martin
(arXiv:2209.09859).
functions that are known to have connections to combinatorics, algebraic
geometry and representation theory. Due to work of Corteel, Mandelshtam
and Williams, it is known that they are related to the asymmetric simple
exclusion process (ASEP) on a ring.
The modified Macdonald polynomials are obtained from the Macdonald
polynomials using an operation called plethysm. It is natural to ask
whether the modified Macdonald polynomials are related to some other
particle system. In this talk, we answer this question in the
affirmative via a multispecies totally asymmetric zero-range process
(TAZRP). We also present a Markov process on tableaux that projects to
the TAZRP and derive formulas for stationary probabilities and certain
correlations. We also prove a remarkable symmetry property for local
correlations.
This is joint work with Olya Mandelshtam and James Martin
(arXiv:2209.09859).