Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz


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.

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  • 09 August 2019

    This is a correction to Theorems 7.3 and 8.12 in [1].

  • 09 August 2019

    This is a correction to Theorems 7.3 and 8.12 in [1].


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Correspondence to Ivan Corwin.

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Communicated by H. Spohn

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Borodin, A., Corwin, I., Petrov, L. et al. Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz. Commun. Math. Phys. 339, 1167–1245 (2015). https://doi.org/10.1007/s00220-015-2424-7

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