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arXiv · 1407.8534

Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz

Alexei Borodin·Ivan Corwin·Leonid Petrov·Tomohiro Sasamoto
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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 which 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 which 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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Alexei Borodin, Ivan Corwin, Leonid Petrov, Tomohiro Sasamoto. 2014-07-31. Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz. https://doi.org/10.1007/s00220-015-2424-7

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