We develop spectral theory for the $q$-Hahn stochastic particle system introduced recently by Povolotsky. That is, we establish a Plancherel type isomorphism result which implies completeness and biorthogonality statements for the Bethe ansatz eigenfunctions of the system. Owing to a Markov duality with the $q$-Hahn TASEP (a discrete-time generalization of TASEP with particles’ jump distribution being the orthogonality weight for the classical $q$-Hahn orthogonal polynomials), we write down moment formulas which characterize the fixed time distribution of the $q$-Hahn TASEP with general initial data.
The Bethe ansatz eigenfunctions of the $q$-Hahn system degenerate into eigenfunctions of other (not necessarily stochastic) interacting particle systems solvable by the coordinate Bethe ansatz. This includes the ASEP, the (asymmetric) six-vertex model, and the Heisenberg XXZ spin chain (all models are on the infinite lattice). In this way, each of the latter systems possesses a spectral theory, too. In particular, biorthogonality of the ASEP eigenfunctions which follows from the corresponding $q$-Hahn statement implies symmetrization identities of Tracy and Widom (for ASEP with either step or step Bernoulli initial configuration) as corollaries. Another degeneration takes the $q$-Hahn system to the $q$-Boson particle system (dual to $q$-TASEP) studied in detail in our previous paper (2013).
Thus, at the spectral theory level we unify two discrete-space regularizations of the Kardar-Parisi-Zhang equation / stochastic heat equation, namely, $q$-TASEP and ASEP.