Spectral theory for interacting particle systems solvable by coordinate Bethe ansatz

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.

65 pages, 13 figures; v3-v4: Removed incorrect proof of spatial biorthogonality of ASEP eigenfunctions. The statement is correct due to Tracy-Widom (arXiv:0704.2633) but does not directly follow from the main results of the present work. See Remark 7.5 for details

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Cited by in corpus (22)

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• Stochastic higher spin vertex models on the line
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• The strict-weak lattice polymer
• Clustering and the Three-Point Function
• Directed random polymers via nested contour integrals
• Half-space Macdonald processes
• The $q$-Hahn asymmetric exclusion process
• Half-space stationary Kardar-Parisi-Zhang equation
• Large Deviations For Sticky Brownian Motions
• KPZ equation limit of stochastic higher spin six vertex model
• Integral formulas of ASEP and $q$-TAZRP on a ring
• Completeness of the Bethe Ansatz for an open $q$-boson system with integrable boundary interactions
• Droplet states in quantum XXZ spin systems on general graphs
• Tracy-Widom limit of q-Hahn TASEP
• Observables of coloured stochastic vertex models and their polymer limits
• Observables of stochastic colored vertex models and local relation
• The stochastic telegraph equation limit of the stochastic higher spin six vertex model
• Tracy-Widom asymptotics for a river delta model
• Current statistics in the q-boson zero range process
• Fluctuations of stochastic charged cellular automata
• Nonstationary generalized TASEP in KPZ and jamming regimes