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Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz

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A Correction to this article was published on 09 August 2019

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Abstract

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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Change history

  • 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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Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA, 02139-4307, USA

    Alexei Borodin & Ivan Corwin

  2. Institute for Information Transmission Problems, Bolshoy Karetny per. 19, Moscow, 127994, Russia

    Alexei Borodin & Leonid Petrov

  3. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, 10027, USA

    Ivan Corwin

  4. Clay Mathematics Institute, 10 Memorial Blvd., Suite 902, Providence, RI, 02903, USA

    Ivan Corwin

  5. Institut Henri Poincaré, 11 Rue Pierre et Marie Curie, 75005, Paris, France

    Ivan Corwin

  6. Department of Mathematics, University of Virginia, 141 Cabell Drive, Kerchof Hall, P.O. Box 400137, Charlottesville, VA, 22904-4137, USA

    Leonid Petrov

  7. Department of Mathematics, Northeastern University, 360 Huntington Ave., Boston, MA, 02115, USA

    Leonid Petrov

  8. Department of Physics, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo, 152-8550, Japan

    Tomohiro Sasamoto

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  1. Alexei Borodin
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  2. Ivan Corwin
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  3. Leonid Petrov
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  4. Tomohiro Sasamoto
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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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