Correction to: Spectral Theory for Interacting Particle Systems Solvable by Coordinate Bethe Ansatz

This is a correction to Theorems 7.3 and 8.12 in [1]. These statements claimed to deduce the spatial Plancherel formula (spatial biorthogonality) of the ASEP and XXZ eigenfunctions from the corresponding statements for the eigenfunctions of the q-Hahn system. Such a reduction is wrong. We are grateful to Yier Lin for pointing this out to us.

We have updated the arXiv version of the paper with the necessary corrections [2]. Below is the summary of the issue and the steps we made to correct the presentation of the ASEP and XXZ applications of our results about the q-Hahn eigenfunctions.

1 q-Hahn Spatial Biorthogonality

Recall that the q-Hahn left and right eigenfunctions are given by

$$\begin{aligned} \Psi ^{\ell }_{{\vec z}}({\vec n})&:= \sum _{\sigma \in S(k)}\prod _{1\le B<A\le k} \frac{z_{\sigma (A)}-qz_{\sigma (B)}}{z_{\sigma (A)}-z_{\sigma (B)}}\prod _{j=1}^{k} \left( \frac{1-z_{\sigma (j)}}{1-\nu z_{\sigma (j)}}\right) ^{-n_j}, \\ \Psi ^{r}_{{\vec z}}({\vec n})&:= (-1)^k(1-q)^{k}q^{\frac{k(k-1)}{2}}{\mathfrak {m}}_{q,\nu }({\vec n}) \sum _{\sigma \in S(k)}\prod _{1\le B<A\le k} \frac{z_{\sigma (A)}-q^{-1}z_{\sigma (B)}}{z_{\sigma (A)}-z_{\sigma (B)}}\prod _{j=1}^{k} \left( \frac{1-z_{\sigma (j)}}{1-\nu z_{\sigma (j)}}\right) ^{n_j} \end{aligned}$$

where $\vec n=(n_1\ge \cdots \ge n_k )$. (Here and below we bring only the essential notation from the original paper [1].) Their spatial biorthogonality written in the small contour form reads [1, Corollary 3.13]

$$\begin{aligned} \sum _{\lambda \vdash k}\oint _{\varvec{\gamma }_k}\ldots \oint _{\varvec{\gamma }_k} d\mathsf {m}^{(q)}_\lambda ({\vec w}) \prod _{j=1}^{\ell (\lambda )}\frac{1}{(w_j;q)_{\lambda _j}(\nu w_j;q)_{\lambda _j}} \Psi ^{\ell }_{{\vec w}\circ \lambda }({\vec n}) \Psi ^{r}_{{\vec w}\circ \lambda }(\vec m)=\mathbf {1}_{\vec m={\vec n}}, \end{aligned}$$

(1)

with all integration contours being small positively oriented circles around 1, and where

$$\begin{aligned} d\mathsf {m}^{(q)}_\lambda ({\vec w}):= \frac{(1-q)^{k}(-1)^{k}q^{-\frac{k^2}{2}}}{m_1!m_2!\ldots } \det \left[ \frac{1}{w_iq^{\lambda _i}-w_j}\right] _{i,j=1}^{\ell (\lambda )} \prod _{j=1}^{\ell (\lambda )}w_j^{\lambda _j}q^{\frac{\lambda _j^{2}}{2}}\frac{dw_j}{2\pi \mathbf {i}}. \end{aligned}$$

Here, ${\vec w}=(w_1,\ldots ,w_{\ell (\lambda )})\in \mathbb {C}^{\ell (\lambda )}$, and $m_j$ is the number of components of $\lambda$ equal to j (so that $\lambda =1^{m_1}2^{m_2}\ldots$), and

$$\begin{aligned} {\vec w}\circ \lambda:= & {} (w_1, qw_1,\ldots , q^{\lambda _1-1}w_1,w_2, qw_2,\ldots , q^{\lambda _{2}-1}w_2,\ldots , w_{\lambda _{\ell (\lambda )}},\\&qw_{\lambda _{\ell (\lambda )}},\ldots , q^{\lambda _{\ell (\lambda )}-1}w_{\lambda _{\ell (\lambda )}}) \in \mathbb {C}^{k}. \end{aligned}$$

2 ASEP Spatial Biorthogonality

To obtain the ASEP eigenfunctions from the q-Hahn ones, we set $\nu =1/q=1/\tau$, where $\tau \in (0,1)$ is the ASEP asymmetry parameter:

$$\begin{aligned} \Psi ^{{\mathrm{ASEP}}}_{{\vec z}}(x_1,\ldots ,x_k)= & {} \Psi ^{\ell }_{-{\vec z}}(x_k,\ldots ,x_1)\vert _{q=\nu ^{-1}=\tau },\\ ({\mathcal {R}}\Psi ^{{\mathrm{ASEP}}}_{{\vec z}})(x_1,\ldots ,x_k) \cdot \mathbf{1}_{x_1<\ldots <x_k}= & {} (\tau ^{-1}-1)^{-k} \Psi ^{r}_{-{\vec z}}(x_k,\ldots ,x_1)\vert _{q=\nu ^{-1}=\tau }. \end{aligned}$$

Here, $x_1<\cdots <x_k$ are the ASEP spatial coordinates. The spatial biorthogonality of the ASEP eigenfunctions reads

$$\begin{aligned} \oint _{{\widetilde{\varvec{\gamma }}}_{-1}}\ldots \oint _{{\widetilde{\varvec{\gamma }}}_{-1}} d\mathsf {m}^{(\tau )}_{(1^{k})}({\vec z})\prod _{j=1}^{k}\frac{1-1/\tau }{(1+z_j)(1+z_j/\tau )} \Psi ^{{\mathrm{ASEP}}}_{{\vec z}}({\vec x})({\mathcal {R}}\Psi ^{{\mathrm{ASEP}}}_{{\vec z}})({\vec y})=\mathbf {1}_{{\vec x}={\vec y}}, \end{aligned}$$

(2)

where the integration is performed over sufficiently small positively oriented circles around $-1$. This biorthogonality of the ASEP eigenfunctions follows from the paper by Tracy and Widom [4], as we explain in detail in [2, Proof of Theorem 7.3]. Next we discuss the gap in our original argument.

3 Why (2) Does Not Follow from (1) as Claimed

The “proof” of ASEP spatial biorthogonality given in [1] claimed to deduce (2) by plugging $\nu =1/q$ into (1) before performing the integration. Indeed, identity (2) looks as if one takes the q-Hahn small contour formula (1), removes all terms corresponding to partitions $\lambda \ne (1^k)$, and then plugs in $\nu =1/q$, $q=\tau$. Formula (2) (following from [4]) a posteriori implies that under this specialization, the contribution of all additional terms with $\lambda \ne (1^k)$ vanishes.

First, observe that the substitution $\nu =1/q$ before the integration might change the value of the integral because of the factors of the form $\frac{1}{1-q \nu w_i}$ in the integrand for $\lambda \ne (1^k)$. Before the substitution $\nu =1/q$, the residue at $w_i=(q \nu )^{-1}$ was not picked while after the substitution we have $1-q \nu w_i=1-w_i$, so this factor adds an extra pole inside the integration contour.

With the agreement that the substitution $\nu =1/q$ occurs after the integration, the “proof” of (2) presented in [1] asserted a stronger statement: For each individual $\lambda \ne (1^k)$ and any two permutations $\sigma , \omega \in S(k)$ (coming from $\Psi _{\vec z}^\ell$ and $\Psi _{\vec z}^{r}$, respectively), the corresponding term vanishes after setting $\nu =1/q$. This assertion is wrong.

For example, take $\vec x=(10,9,8,7,6,5)$ and $\vec y=(5,4,3,2,1,0)$. The summand in the integrand in (1) corresponding to $\lambda =(3,2,1)$, and permutations $\sigma =321{,}546$ and $\omega =645{,}123$ has the form (before setting $q=1/\nu =\tau$):

$$\begin{aligned}&\mathrm {const}\cdot \frac{(1-\nu q w_1)^7 (1-\nu q w_2)^3}{(1-w_1)^7 (1-w_2)^3 (1-w_3)}\\&\quad \times \frac{ (q w_1-w_2) \left( q^2 w_1-w_2\right) ^2 \left( q^3 w_1-w_2\right) \left( q^2 w_1-w_3\right) \left( q^3 w_1-w_3\right) (q w_2-w_3) \left( q^2 w_2-w_3\right) }{ (w_1-w_2) (w_1-w_3) (w_2-w_3) (q w_2-w_1)^2 \left( q^2 w_2-w_1\right) (q w_3-w_1) (q w_3-w_2) }\\&\quad \times f_1(w_1)f_2(w_2)f_3(w_3). \end{aligned}$$

Here, $f_1(w_1)$ is independent of $w_2,w_3$ and has no zeroes or poles at $w_1=1$ and $w_1=1/(q\nu )$, and similarly for $f_2(w_2)$ and $f_3(w_3)$. One can check that the residue of this term at $w_3=1$, $w_2=1$, and $w_1=1$ does not vanish when setting $q=1/\nu$. (Note that the result of the integration depends on the order of taking the residues for individual summands due to the presence of the factors of the form $w_i-w_j$ in the denominators. These factors cancel out after summing over all permutations $\sigma ,\omega$, and each summand indexed by $\lambda$ is independent of the order of integration because the result of the summation is a function symmetric in the $w_i$’s.)

Let us mention another (possibly related) subtlety in the spatial biorthogonality of the ASEP eigenfunctions as compared to the general q-Hahn case. Namely, in the q-Hahn situation the contribution of individual permutations coming from the eigenfunctions vanishes, while in the ASEP case this is not the case (see [2, Remark 7.6] for details). The proof of the ASEP statement in [4] employs nontrivial combinatorics to determine cancellations of specific combinations of permutations.

4 Corrections We Made in the New Version [2] Compared to the Published Version [1]

We have replaced the incorrect “proof” of Theorem 7.3 (spatial biorthogonality of the ASEP eigenfunctions) by its derivation from the earlier result of Tracy and Widom [4]. We have also removed Theorem 8.12 which claimed a spatial biorthogonality statement of the XXZ eigenfunctions based on a similar incorrect direct substitution $\nu ={\varvec{\theta }}$.

5 The Same Gap in [3]

The claim similar to (1) but with more general $\nu =q^{-I}$, where I is an arbitrary positive integer, is made in [3, Appendix A] (by a subset of the current authors). When $I=1$, this identity is correct, but does not follow from the general $\nu \in (0,1)$ formulas (as explained above). Moreover, for $I\ge 2$ the claimed orthogonality does not seem to hold as stated. A separate erratum will be prepared to address the issues in the work [3].