Finn–Vanicat Conjecture 4.1: the $m\to\infty$ limit of Koornwinder matrix-product states and an effective exact formula for the current cumulant generating function of the open ASEP

The open asymmetric simple exclusion process (ASEP) — a chain of $N$ sites coupled to particle reservoirs at both ends — is the paradigmatic exactly tractable model of a boundary-driven nonequilibrium steady state, and the full counting statistics of the current exchanged with a reservoir — encoded in the scaled cumulant generating function $E(\mu)$, the largest eigenvalue $\Lambda_0$ of the current-deformed Markov matrix $M(\xi)$ at fugacity $\xi=e^{\mu}$ — is its central still-unresolved observable at finite size and generic parameters. Finn and Vanicat (2017) solved, in matrix-product form, boundary quantum Knizhnik–Zamolodchikov (qKZ) equations with an additional deformation, at the constraint $\xi=s^{m}$, and proved that the normalization $Z^{(m)}$ of their twice-deformed ground state is the symmetric Koornwinder polynomial $P_{(m^{N})}$ with rectangular partition. They conjectured (Conjecture 4.1) that, as $m$ tends to infinity with $s=\xi^{1/m}$, the normalized states converge to a regular vector $|\Psi_{0}\rangle$ and $\frac{\ln\xi}{m}\ln Z^{(m)}$ converges to a regular function $F_{0}$, from which the current statistics follows through $E(\mu)=\frac{p-q}{2}\,\frac{\partial^{2}F_{0}}{\partial x_i^{2}}(\mathbf{1};e^{\mu})$ evaluated at $\mathbf{1}=(1,\dots,1)$. The conjecture is supported only by numerics up to three sites; the characterization of $F_{0}$ by a nonlinear functional equation has no known closed-form or constructive solution, so the formula for $E(\mu)$ remains formal. The problem is to prove or disprove the conjectured convergence and — should it hold — to determine $F_{0}$ effectively, thereby making the exact current statistics of the finite open ASEP computable non-perturbatively.

Background

The asymmetric simple exclusion process (ASEP) is a stochastic lattice gas on a chain of $N$ sites in which particles hop right and left with rates $p$ and $q$, with at most one particle per site (exclusion). In the open ASEP the chain is coupled to two reservoirs: particles are injected at the left end with rate $\alpha$ and removed with rate $\gamma$; at the right end, particles are extracted with rate $\beta$ and injected with rate $\delta$ (the boundary-rate convention of the source paper). The system relaxes to a nonequilibrium steady state carrying a macroscopic current, and its exact stationary measure is given by a matrix product ansatz, in which the stationary weight of a configuration is a product of non-commuting matrices obeying quadratic algebraic relations (Derrida, Evans, Hakim and Pasquier, 1993).

Because the current is the order parameter of a boundary-driven nonequilibrium system, its fluctuation statistics is a central observable. Writing $Q_T$ for the algebraic number of particles exchanged between the left reservoir and the chain up to time $T$, the probability of an atypical flux obeys $P(Q_T/T=j)\sim e^{-T G(j)}$ at large $T$, where $G(j)$ is the current large deviation function, the Legendre transform of the scaled cumulant generating function $E(\mu)$. The latter is obtained by deforming the Markov generator by a fugacity $\xi=e^{\mu}$ counting each jump across the left boundary, $M(\xi)=\sqrt{\alpha\gamma}\,B_1(\xi)+\sum_{i=1}^{N-1}\sqrt{pq}\,w_{i,i+1}+\sqrt{\beta\delta}\,B_N$ (the local bulk matrix $w$ and boundary matrices $B_1(\xi),B_N$ encode the hopping and reservoir rates; $M(1)$ is the true stochastic generator), and taking its largest eigenvalue: $E(\mu)=\Lambda_0(e^{\mu})$, with $\Lambda_0(1)=0$. The Gallavotti–Cohen fluctuation symmetry relates large positive and negative currents; for the open ASEP it states that $\Lambda_0$ is invariant under $\xi\to\xi'=(\gamma\delta/\alpha\beta)(q/p)^{N-1}\,\xi^{-1}$.

What is known about $E(\mu)$: for the periodic ASEP the current large deviation function is exactly known (Derrida and Lebowitz, 1998); for the open TASEP (the totally asymmetric case $q=0$) an exact formula for the statistics of the current exists (Lazarescu and Mallick, 2011); for the open ASEP in the thermodynamic limit, de Gier and Essler (2011) used the Bethe ansatz — the exact method that represents eigenvectors as superpositions of plane waves, here applied to the deformed stochastic generator — to obtain the current statistics at the first site in the low- and high-density phases, and the exact diffusion constant in the steady state is known for all $N$ (Derrida, Evans and Mallick, 1995); at finite size, the perturbative matrix ansatz (Gorissen, Lazarescu, Mallick and Vanderzande, 2012; Lazarescu, 2013) yields every cumulant of the current order by order in the expansion about equilibrium $\xi=1$, at the price of solving nonlinear implicit equations at each order, and Lazarescu and Pasquier (2014) constructed for the deformed open ASEP a Baxter $Q$-operator — an auxiliary operator commuting with the deformed transfer matrix, a standard route to its spectrum — and the associated $t$-$Q$ relations. A closed, non-perturbative expression for $E(\mu)$ at finite $N$ and generic parameters is nevertheless not known, a gap explicitly recorded in Lazarescu's 2015 review of current fluctuations in bulk-driven lattice gases.

Finn and Vanicat (2017) connected this problem to the algebraic combinatorics of the model. The stationary states of multispecies open ASEP are known to be tied to multivariate Koornwinder polynomials (Cantini, 2015; Cantini, Garbali, de Gier and Wheeler, 2016) — the Macdonald polynomials of affine type $C$, a multiparameter family of symmetric Laurent polynomials generalizing the one-variable Askey–Wilson polynomials (Cantini, de Gier and Wheeler, 2015, established the analogous Macdonald result for the periodic chain). The symmetric Koornwinder polynomial $P_\lambda$ with partition $\lambda$ is the unique $W_0$-invariant element of its span with unit leading monomial, where $W_0$ is the finite Weyl group of type $C_N$ generated by the transpositions $s_i:x_i\leftrightarrow x_{i+1}$ ($1\le i\le N-1$) and the inversion $s_N:x_N\mapsto x_N^{-1}$, i.e. the signed permutations of the variables $x_1,\dots,x_N$; it is an eigenfunction, with eigenvalue $d_\lambda$, of the conserved difference operator

$$D=\sum_{i=1}^{N}\Big[g_i(\mathbf{x})(T_{s,i}-1)+g_i(\mathbf{x}^{-1})(T_{s,i}^{-1}-1)\Big],$$

where $T_{s,i}$ is the shift $x_i\mapsto s\,x_i$ and the $g_i$ are explicit rational functions of $\mathbf{x}$ depending on the Koornwinder parameters. The ASEP rates fix these parameters through the dictionary $t=p/q$, $t_0=\alpha/\gamma$, $t_N=\beta/\delta$, with $u_0,u_N$ determined by $u_0^{1/2}-u_0^{-1/2}=(p-q+\gamma-\alpha)/\sqrt{\alpha\gamma}$ and $u_N^{1/2}-u_N^{-1/2}=(p-q+\delta-\beta)/\sqrt{\beta\delta}$. Finn and Vanicat introduced a further (second) deformation of the current-deformed ground state by solving boundary quantum Knizhnik–Zamolodchikov (qKZ) equations — difference equations, with step $s$ in the spectral variables, satisfied by correlation functions of integrable models — in matrix-product form in an infinite-dimensional Fock-space representation of the boundary Yang–Baxter algebra, the algebra generated by the bulk and boundary matrices of the underlying integrable open-chain structure. At the constraint $\xi=s^m$ ($m$ a positive integer) they proved (Theorem 3.14) that the normalization $Z^{(m)}(\mathbf{x};s)=\langle 1|\Psi^{(m)}(\mathbf{x};s)\rangle$, i.e. the sum of components of the twice-deformed ground state, equals exactly the symmetric Koornwinder polynomial $P_{(m^N)}(\mathbf{x})$ with rectangular partition $(m,\dots,m)$.

The link to current statistics arises in the simultaneous limit $m\to\infty$, $s=\xi^{1/m}\to1$, a quasi-classical-type degeneration of the qKZ equations (a limit in which the step parameter tends to $1$ and difference equations degenerate towards differential equations), analogous to the quasi-classical asymptotics of Knizhnik–Zamolodchikov-type equations studied by Reshetikhin (1992) and Reshetikhin and Varchenko (1994). Finn and Vanicat conjecture (Conjecture 4.1), on the basis of numerics up to $N=3$ sites, that in this limit the normalized states converge to a vector $|\Psi_0(\mathbf{x};\xi)\rangle$ regular in $\mathbf{x}$ and that $\frac{\ln\xi}{m}\ln Z^{(m)}(\mathbf{x};\,s=\xi^{1/m})$ converges to a function $F_0(\mathbf{x};\xi)$ regular in $\mathbf{x}$. Treating the conjecture as fact, they then derive (Propositions 4.2–4.5): $F_0$ is $W_0$-invariant with the Gallavotti–Cohen symmetry on its gradient; $|\Psi_0\rangle$ is an eigenvector of the qKZ scattering matrices at $s=1$ with eigenvalue $\exp(x_i\,\partial F_0/\partial x_i)$; the ground-state relation of the deformed Markov matrix,

$$M(\xi)\,|\Psi_0(\mathbf{1};\xi)\rangle=\frac{p-q}{2}\,\frac{\partial^2 F_0}{\partial x_i^2}(\mathbf{1};\xi)\,|\Psi_0(\mathbf{1};\xi)\rangle,$$

which yields the generating function of the cumulants of the current as $E(\mu)=\Lambda_0(e^{\mu})=\frac{p-q}{2}\frac{\partial^2 F_0}{\partial x_i^2}(\mathbf{1};\,e^{\mu})$ with $\mathbf{1}=(1,\dots,1)$ (the right-hand side being independent of $i$ by $W_0$ invariance); and a characterization of $F_0$ as the solution of the nonlinear functional equation

$$\sum_{i=1}^{N}\Big[g_i(\mathbf{x};1)\big(e^{x_i\,\partial F_0/\partial x_i}-1\big)+g_i(\mathbf{x}^{-1};1)\big(e^{-x_i\,\partial F_0/\partial x_i}-1\big)\Big]=d_0(\xi),$$

obtained as the $m\to\infty$ limit of the Koornwinder eigenvalue equation $DP_{(m^N)}=d_{(m^N)}P_{(m^N)}$, with $d_0(\xi)=\frac{1-t^N}{1-t}(\xi-1)\big(t_0t_Nt^{N-1}-\xi^{-1}\big)$. The authors stress that in practice this does not yet compute the current cumulants, because no closed expression for $F_0$ is known. They also record two adjacent questions: whether the characterization of $F_0$ can be reformulated through Baxter $t$-$Q$ relations (with the stationary normalization of inhomogeneous discrete-time exclusion processes, which satisfies such a difference equation, and the Q-operator of Lazarescu and Pasquier as precedents), and whether the TASEP limit $t\to\infty$ (i.e. $q\to0$) of the construction simplifies it.

Problem Statement

For the open ASEP with $N$ sites, generic bulk rates $p,q>0$ with $p\ne q$ and generic boundary rates $\alpha,\beta,\gamma,\delta>0$, let $M(\xi)$ be the current-counting deformed Markov matrix above, and let $|\Psi^{(m)}(\mathbf{x};s)\rangle$ be Finn and Vanicat's twice-deformed ground state: the matrix-product solution of the boundary qKZ equations at the constraint $\xi=s^m$ ($m$ a positive integer), with normalization $Z^{(m)}(\mathbf{x};s)=\langle1|\Psi^{(m)}(\mathbf{x};s)\rangle=P_{(m^N)}(\mathbf{x})$, the symmetric Koornwinder polynomial with rectangular partition $(m,\dots,m)$ in the ASEP parameter dictionary ($t=p/q$, $t_0=\alpha/\gamma$, $t_N=\beta/\delta$, $u_0,u_N$ as above).

(i) Prove, or disprove, Conjecture 4.1 of Finn and Vanicat: as $m\to\infty$ with $s=\xi^{1/m}$,

$$\lim_{m\to\infty}\frac{|\Psi^{(m)}(\mathbf{x};\,s=\xi^{1/m})\rangle}{Z^{(m)}(\mathbf{x};\,s=\xi^{1/m})}=|\Psi_0(\mathbf{x};\xi)\rangle,$$ $$\lim_{m\to\infty}\frac{\ln\xi}{m}\,\ln Z^{(m)}(\mathbf{x};\,s=\xi^{1/m})=\lim_{m\to\infty}\frac{\ln\xi}{m}\,\ln P_{(m^N)}(\mathbf{x};\,s=\xi^{1/m})=F_0(\mathbf{x};\xi),$$

with $|\Psi_0(\mathbf{x};\xi)\rangle$ and $F_0(\mathbf{x};\xi)$ regular functions of $\mathbf{x}=(x_1,\dots,x_N)$ (regular meaning analytic, i.e. holomorphic, in $\mathbf{x}$).

(ii) Assuming the convergence in (i), determine $F_0$ effectively for general $N$ and generic parameters, as a regular $W_0$-invariant solution of the functional equation

$$\sum_{i=1}^{N}\Big[g_i(\mathbf{x};1)\big(e^{x_i\,\partial F_0/\partial x_i}-1\big)+g_i(\mathbf{x}^{-1};1)\big(e^{-x_i\,\partial F_0/\partial x_i}-1\big)\Big]=d_0(\xi),$$

with $d_0(\xi)=\frac{1-t^N}{1-t}(\xi-1)(t_0t_Nt^{N-1}-\xi^{-1})$ and $g_i$ the rational coefficients of the Koornwinder difference operator $D$ defined in the background, so that the relation

$$E(\mu)=\Lambda_0(e^{\mu})=\frac{p-q}{2}\,\frac{\partial^2F_0}{\partial x_i^2}(\mathbf{1};\,e^{\mu})$$

becomes an effective, non-perturbative, exact formula for the scaled cumulant generating function of the current of the finite open ASEP, from which all current cumulants and the current large deviation function follow.

The core open question is the convergence in (i). A disproof of (i) — non-existence or non-regularity of the limits for an admissible parameter set — settles the conjecture negatively and, with it, the conditional status of Propositions 4.2–4.5 of the source; part (ii) then becomes void, since $F_0$ and the functional equation of Proposition 4.5 are defined only under the conjecture. Part (ii) reflects the source's remark that, even granting the conjecture, the formula for $E(\mu)$ 'does not help to compute the cumulants of the current because a closed expression for $F_0$ is missing'; it is a follow-on goal conditional on (i), and the source states the functional equation only as a characterization of $F_0$, attaching no uniqueness claim and no completion criterion to it. Two sub-directions flagged by the authors fall within the same objective but do not separately bound completion: whether the functional equation for $F_0$ can be recast as a Baxter $t$-$Q$ relation, and whether the TASEP limit $t\to\infty$ (i.e. $q\to0$) of the construction simplifies it.

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