I will focus on two interacting particle systems with inhomogeneous rates on the finite ring with site-dependent rates. The first is the Totally Asymmetric Zero Range Process (TAZRP) and the second is the PushTASEP, where TASEP stands for the Totally Asymmetric Simple Exclusion Process. In both cases, I will present results which say that the distribution of the path of the process restricted to certain intervals is unchanged under permutations of the rates. Our proofs build on Weber’s theorem for exponential queues. I will aim to keep the talk self-contained.

This is joint work with O. Mandelshtam and J. Martin (arXiv:2209.09859, Math. Z., to appear) and with J. Martin (arXiv:2310.09740).