SPECTRAL THEORY FOR INTERACTING PARTICLE SYSTEMS ...

SPECTRAL THEORY FOR INTERACTING PARTICLE SYSTEMS SOLVABLE BY COORDINATE BETHE ANSATZ

arXiv:1407.8534v1 [math-ph] 31 Jul 2014

ALEXEI BORODIN, IVAN CORWIN, LEONID PETROV, AND TOMOHIRO SASAMOTO

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.

Contents 1. Introduction 2. Definition of eigenfunctions 3. Plancherel formulas 4. Spectral biorthogonality of eigenfunctions 5. The q-Hahn system and coordinate Bethe ansatz 6. Conjugated q-Hahn operator 7. Application to ASEP 8. Application to six-vertex model and XXZ spin chain 9. Appendix. Further degenerations References

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1. Introduction 1.1. Main results for the q-Hahn system eigenfunctions. The q-Hahn system introduced by Povolotsky [48] is a discrete-time stochastic Markov dynamics on k-particle configurations on Z (where k ≥ 1 is arbitrary) in which multiple particles at a site are allowed (in fact, it is a totally asymmetric zero-range process, see [38] for a general background). At each step of the q-Hahn system dynamics, independently at every occupied side i ∈ Z with yi ≥ 1 particles, one randomly selects si ∈ {0, 1, . . . , yi } particles according to the probability distribution ϕq,µ,ν (si | yi ) = µsi

(ν/µ; q)si (µ; q)yi −si (q; q)yi , (ν; q)yi (q; q)si (q; q)yi −si

1

(a; q)n :=

n Y

(1 − aq j−1 ),

j=1

SPECTRAL THEORY FOR INTERACTING PARTICLE SYSTEMS

2

where 0 < q < 1 and 0 ≤ ν ≤ µ < 1 are three parameters of the model. These selected si particles are immediately moved to the left (i.e., to site i − 1). This update occurs in parallel for each site. See Fig. 6 in §5.2, left panel. Configurations can be encoded by vectors ~n = (n1 ≥ . . . ≥ nk ), ni ∈ Z, where ni is the position of the i-th particle from the right. We denote by Wk the space of all such vectors. Therefore, the backward bwd of the q-Hahn stochastic process acts on the space of compactly Markov transition operator Hq,µ,ν supported functions in the spatial variables ~n. We denote the latter space by W k . The left and right bwd are, respectively1 eigenfunctions of the operator Hq,µ,ν  k  X Y 1 − zσ(j) −nj zσ(A) − qzσ(B) Y ℓ ; Ψ~z (~n) = zσ(A) − zσ(B) 1 − νzσ(j) j=1 σ∈S(k) 1≤B