2014/07/31 by Alexei Borodin, Ivan Corwin, Leonid Petrov +1
Physics and Astronomy · Mathematics · #math-ph #cond-mat.stat-mech #math.MP #math.PR #math.QA #math.RT
paper · pdf · doi:10.1007/s00220-015-2424-7
published as Comm. Math. Phys. 339 (2015), no. 3, 1167--1245 · 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
arxiv created 2018/12/19 · arxiv updated 2018/12/21
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.