Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz

• Published in: Communications in mathematical physics 2015-11, Vol.339 (3), p.1167-1245
• Main Authors: Borodin, Alexei, Corwin, Ivan, Petrov, Leonid, Sasamoto, Tomohiro
• Format: Article
• Language: English
• Subjects: Classical and Quantum Gravitation; Complex Systems; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Relativity Theory; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-015-2424-7

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.