## Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz

##### Author(s)

Borodin, Alexei; Corwin, Ivan; Petrov, Leonid; Sasamoto, Tomohiro
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We develop spectral theory for the q-Hahn stochastic particle system introduced recently by Povolotsky. That is, we establish a Plancherel type isomorphism result that 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 that 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.

##### Date issued

2015-07##### Department

Massachusetts Institute of Technology. Department of Mathematics##### Journal

Communications in Mathematical Physics

##### Publisher

Springer Berlin Heidelberg

##### Citation

Borodin, Alexei et al. “Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz.” Communications in Mathematical Physics 339.3 (2015): 1167–1245.

Version: Author's final manuscript

##### ISSN

0010-3616

1432-0916