Communications in Mathematical Physics

, Volume 339, Issue 3, pp 1167–1245 | Cite as

Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz

  • Alexei Borodin
  • Ivan CorwinEmail author
  • Leonid Petrov
  • Tomohiro Sasamoto


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.


Corwin Laurent Polynomial Bilinear Pairing Interact Particle System Totally Asymmetric Simple Exclusion Process 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Alexei Borodin
    • 1
    • 2
  • Ivan Corwin
    • 1
    • 3
    • 4
    • 5
    Email author
  • Leonid Petrov
    • 2
    • 6
    • 7
  • Tomohiro Sasamoto
    • 8
  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Institute for Information Transmission ProblemsMoscowRussia
  3. 3.Department of MathematicsColumbia UniversityNew YorkUSA
  4. 4.Clay Mathematics InstituteProvidenceUSA
  5. 5.Institut Henri PoincaréParisFrance
  6. 6.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA
  7. 7.Department of MathematicsNortheastern UniversityBostonUSA
  8. 8.Department of PhysicsTokyo Institute of TechnologyTokyoJapan

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