Interacting particle systems form an important class of toy models in
statistical mechanics. Some of these models belong to the so-called integrable class and many of these in turn have rich combinatorial properties. The most famous of these is the asymmetric simple exclusion process (ASEP), but there are many others. Understanding the combinatorics of these models will enable us to compute quantities useful to physicists, and is therefore of practical importance. One signature of the combinatorial nature of the model is that the partition function is a symmetric polynomial.
After explaining the setup, I will first review known examples of combinatorial interacting particle systems, with a special focus on finite state models. I will then focus on two new models where the Macdonald polynomials seem to play an important role. The first is the multispecies totally asymmetric zero-range process (mTAZRP) and the second is the multispecies totally asymmetric long-range exclusion process (mTALREP). I will end with open problems and possible directions for future work.
Some of what I present is ongoing work with Olya Mandelshtam, James Martin, Omer Angel and Gideon Amir.