Borodin, Corwin, Petrov and Sasamoto (2015) established the spectral (Plancherel) theory of the coordinate Bethe ansatz eigenfunctions of the qq-Hahn interacting particle system, including completeness of the Bethe ansatz and a spectral biorthogonality theorem for generic spectral variables. They conjectured, as the formal identity (4.14) of their paper, that when the spectral variables specialize to qq-strings indexed by partitions of the particle number kk, the eigenfunction biorthogonality degenerates to a compact determinant-of-deltas identity; they left the admissible test-function classes and integration contours unspecified and verified only the single case k=2k=2 with the two-element string. This problem asks for a rigorous formulation and a proof (or refutation) of this degenerate spectral biorthogonality for every kk, all pairs of string partitions, and the full stochastic parameter domain. No resolution of this conjecture is known in the literature to date.

Background

Integrable probability studies stochastic interacting particle systems that possess enough algebraic structure to be analyzed exactly. A basic example is TASEP (totally asymmetric simple exclusion process), in which particles on the integer lattice Z\mathbb{Z} jump one step to the right into vacant sites. The qq-Hahn TASEP, introduced by Povolotsky (2013), is a three-parameter discrete-time generalization: at each time step every particle jumps to the right by a random number of sites drawn from the qq-Hahn (qq-beta-binomial) distribution with parameters 0<q<10<q<1 and 0≤ν≤μ<10\le\nu\le\mu<1, where the number of consecutive vacant sites ahead of the particle enters the weight. It contains discrete-time qq-TASEP and other exactly solvable systems as degenerations, and its zero-range dual (the qq-Hahn Boson process of Corwin, a particle system on site-occupation numbers in which particles hop right by qq-Hahn-distributed amounts) is diagonalized together with it by the coordinate Bethe ansatz - the classical method, going back to Bethe (1931), in which eigenfunctions of the transition operator are written as sums over permutations of kk single-particle factors, with two-body scattering encoded by rational factors.

Borodin, Corwin, Petrov and Sasamoto (BCPS, Communications in Mathematical Physics 339 (2015) 1167-1245) developed a complete spectral theory for these eigenfunctions. Particle configurations are ordered kk-tuples n⃗=(n1≥…≥nk)\vec n=(n_1\ge\ldots\ge n_k) in Z\mathbb{Z} (a discrete Weyl chamber Wk\mathcal W_k). The right eigenfunctions Ψz⃗r(n⃗)\Psi^{r}_{\vec z}(\vec n) and left eigenfunctions Ψz⃗ℓ(n⃗)\Psi^{\ell}_{\vec z}(\vec n) of the qq-Hahn transition operator are symmetric functions of kk spectral variables z⃗=(z1,…,zk)\vec z=(z_1,\ldots,z_k), given as sums over the symmetric group S(k)S(k) of products of qq-Hahn building blocks with two-body factors (zA−qzB)/(zA−zB)(z_A-qz_B)/(z_A-z_B); as functions of z⃗\vec z they are Laurent polynomials in the transformed variables Ξ(zj)=(1−zj)/(1−νzj)\Xi(z_j)=(1-z_j)/(1-\nu z_j). BCPS proved that the transform sending a compactly supported function of n⃗\vec n to a symmetric Laurent polynomial in z⃗\vec z (integration of the left eigenfunctions against a Plancherel-type spectral measure) is an isomorphism - a spatial Plancherel formula analogous to Fourier inversion - together with the dual spectral Plancherel formula, completeness of the Bethe ansatz, and a spectral biorthogonality theorem (their Theorem 4.3) for generic, pairwise distinct spectral variables. These results produce moment formulas for the qq-Hahn TASEP with arbitrary initial conditions and degenerate to parallel statements for ASEP (the asymmetric simple exclusion process, the continuous-time nearest-neighbour analogue of TASEP), the stochastic six-vertex model (a random lattice-path model in which paths routing through each vertex obey six-vertex weights), the Heisenberg XXZ spin chain (a one-dimensional quantum spin model with anisotropic nearest-neighbour interaction), and the qq-Boson system (a bosonic lattice model with qq-deformed commutation relations).

For a partition λ=(λ1≥λ2≥…)\lambda=(\lambda_1\ge\lambda_2\ge\ldots) of kk, written λ⊢k\lambda\vdash k, and independent variables z⃗′=(z1′,…,zℓ(λ)′)\vec z'=(z'_1,\ldots,z'_{\ell(\lambda)}), the string z⃗=z⃗′∘λ\vec z=\vec z'\circ\lambda is the kk-tuple (z1′,qz1′,…,qλ1−1z1′,  z2′,qz2′,…,qλ2−1z2′,  …)(z'_1,qz'_1,\ldots,q^{\lambda_1-1}z'_1,\;z'_2,qz'_2,\ldots,q^{\lambda_2-1}z'_2,\;\ldots). Eigenfunctions with string-specialized spectral variables are exactly those appearing in the completeness expansion of BCPS (their Corollary 3.10 sums over all partitions λ⊢k\lambda\vdash k), and the analogous 'string hypothesis' states play an essential role in the physics of exactly solved models: for the one-dimensional delta Bose gas (particles on the line interacting through pointwise attractive delta-function forces) in the attractive regime, Bethe wave functions whose rapidities form strings are required for completeness, and their degenerate orthogonality relations have been studied in the physics literature.

BCPS conjectured (identity (4.14) of their paper, stated below in the problem) that the spectral biorthogonality degenerates to a compact determinant-of-deltas formula under string specializations. The conjecture is only formal: the authors state that 'it is not clear which test functions or integration contours should be used'; they observe that the case of two different partitions vanishes (by the ideas of their Section 4.3); and they verify the single instance k=2k=2, λ=κ=(2)\lambda=\kappa=(2), where the delta-determinant collapses to a single Dirac delta δ(z′−w′)\delta(z'-w') and the check was performed on Laurent-polynomial test functions in Ξ(z′)\Xi(z') with small positively oriented circular contours around the point 11.

Problem Statement

For the q-Hahn system with parameters 0<q<10<q<1 and 0≤ν<10\le\nu<1 (with 0≤ν≤μ<10\le\nu\le\mu<1 wherever the q-Hahn Markov operator is invoked), let Ψz⃗r(n⃗)\Psi^{r}_{\vec z}(\vec n) and Ψz⃗ℓ(n⃗)\Psi^{\ell}_{\vec z}(\vec n) be the right and left coordinate Bethe eigenfunctions of Borodin-Corwin-Petrov-Sasamoto (their formulas (2.14) and (2.11)), viewed as functions of the kk-particle configuration n⃗=(n1≥…≥nk)\vec n=(n_1\ge\ldots\ge n_k) in the discrete Weyl chamber Wk\mathcal W_k and of the spectral variables z⃗∈Ck\vec z\in\mathbb{C}^{k}. For a partition λ⊢k\lambda\vdash k define the string z⃗=z⃗′∘λ=(z1′,qz1′,…,qλ1−1z1′,  z2′,qz2′,…,  qλℓ(λ)−1zℓ(λ)′)\vec z=\vec z'\circ\lambda=(z'_1,qz'_1,\ldots,q^{\lambda_1-1}z'_1,\;z'_2,qz'_2,\ldots,\;q^{\lambda_{\ell(\lambda)}-1}z'_{\ell(\lambda)}), and likewise w⃗=w⃗′∘κ\vec w=\vec w'\circ\kappa. Let ⟨f,g⟩Wk=∑n⃗∈Wkf(n⃗)g(n⃗)\langle f,g\rangle_{\mathcal W_k}=\sum_{\vec n\in\mathcal W_k}f(\vec n)g(\vec n) denote the spatial bilinear pairing, let V(z⃗)=∏A<B(zA−zB)V(\vec z)=\prod_{A<B}(z_A-z_B) denote the Vandermonde determinant, and let ∏A≠B∼(zA−qzB)\prod\nolimits^{\sim}_{A\ne B}(z_A-qz_B) denote the product over all pairs A≠BA\ne B of string variables with the factors that vanish identically because of the string constraint omitted. The conjecture of BCPS (identity (4.14)) is the formal identity

⟨Ψz⃗′∘λr,Ψw⃗′∘κℓ⟩Wk V(z⃗′∘λ) V(w⃗′∘κ)=1λ=κ (−1)k(k−1)/2∏j=1k(1−zj)(1−νzj)∏A≠B∼(zA−qzB)det⁡[δ(zi−wj)]i,j=1k,\bigl\langle \Psi^{r}_{\vec z'\circ\lambda},\Psi^{\ell}_{\vec w'\circ\kappa}\bigr\rangle_{\mathcal W_k}\,V(\vec z'\circ\lambda)\,V(\vec w'\circ\kappa)=\mathbf 1_{\lambda=\kappa}\,(-1)^{k(k-1)/2}\prod_{j=1}^{k}(1-z_j)(1-\nu z_j)\prod\nolimits^{\sim}_{A\ne B}(z_A-qz_B)\det\bigl[\delta(z_i-w_j)\bigr]_{i,j=1}^{k},

where on the right-hand side the products over jj and over A≠BA\ne B run over the string variables z1,…,zkz_1,\ldots,z_k of z⃗=z⃗′∘λ\vec z=\vec z'\circ\lambda, and the Dirac-delta determinant is to be interpreted in the independent spectral variables z⃗′\vec z' and w⃗′\vec w' (for example, for k=2k=2 and λ=κ=(2)\lambda=\kappa=(2) it reduces to the single distribution δ(z′−w′)\delta(z'-w')).

The task is to settle this conjecture: (1) formulate it rigorously, by specifying a class of admissible test functions and integration contours in the independent variables z⃗′\vec z' and w⃗′\vec w' (in the sense of the integral formulation of the non-degenerate Theorem 4.3 of BCPS) under which both sides of the identity are well-defined; and (2) prove, or refute, the resulting statement for every particle number k≥1k\ge 1 and every pair of partitions λ,κ⊢k\lambda,\kappa\vdash k, over the full parameter domain 0<q<10<q<1, 0≤ν<10\le\nu<1.

A complete solution must be consistent with the established special cases: it must reproduce the BCPS verification for k=2k=2, λ=κ=(2)\lambda=\kappa=(2) (Laurent-polynomial test functions in the transformed variables Ξ(z′)=(1−z′)/(1−νz′)\Xi(z')=(1-z')/(1-\nu z'), small positively oriented contours around 11), the vanishing of the pairing when λ≠κ\lambda\ne\kappa, and the non-degenerate spectral biorthogonality (Theorem 4.3 of BCPS) when λ=κ=(1k)\lambda=\kappa=(1^{k}). Results covering only bounded kk, a proper subset of partitions, or ν=0\nu=0 constitute partial progress, not a resolution. A natural corollary, not required, is the corresponding degenerate version of the spectral biorthogonality in the Ξ\Xi-variables (Theorem 4.4 of BCPS) obtained via the change of variables Ξ\Xi.

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