Duality Relations for the Periodic ASEP Conditioned on a Low Current

Abstract

We consider the asymmetric simple exclusion process (ASEP) on a finite lattice with periodic boundary conditions, conditioned to carry an atypically low current. For an infinite discrete set of currents, parametrized by the driving strength $s_K$, $K \ge 1$, we prove duality relations which arise from the quantum algebra $U_q[\mathfrak {gl}(2)]$ symmetry of the generator of the process with reflecting boundary conditions. Using these duality relations we prove on microscopic level a travelling-wave property of the conditioned process for a family of shock-antishock measures for $N>K$ particles: If the initial measure is a member of this family with K microscopic shocks at positions $(x_1,\dots ,x_K)$, then the measure at any time $t>0$ of the process with driving strength $s_K$ is a convex combination of such measures with shocks at positions $(y_1,\dots ,y_K)$, which can be expressed in terms of K-particle transition probabilities of the conditioned ASEP with driving strength $s_N$.

Appendix

We present some details of the proof of Proposition (1) which are not shown in [30] and from which Proposition (1) follows by the similarity transformation (33).

We define $e_L(\cdot ,\cdot ,\cdot ) := h_{L,1}(\cdot ,\cdot ,\cdot )$, see (31). By explicit matrix multiplications one finds from the relations (25) for the bulk operators

$$\begin{aligned} S_k^\pm (q,\alpha ) e_L(\alpha ',q',\beta )= & {} e_L(\alpha ',q',\beta q^{-2}) S_k^\pm (q,\alpha ) \quad 2 \le k \le L-1 \end{aligned}$$

(131)

$$\begin{aligned} e_L(\alpha ',q',\beta ) S_k^\pm (q,\alpha )= & {} S_k^\pm (q,\alpha ) e_L(\alpha ',q',\beta q^{2}) \quad 2 \le k \le L-1 \end{aligned}$$

(132)

and for the boundary operators

$$\begin{aligned} S_1^+(q,\alpha ) e_L(\alpha ',q',\beta )= & {} q^{-1/2} \alpha ^{\frac{1}{2}(L-1)} \left[ (q')^{-1} \sigma ^+_1 \hat{\upsilon }_L - \alpha ' \beta \hat{\upsilon }_1 \sigma ^+_L \right] q^{-S^z}\qquad \end{aligned}$$

(133)

$$\begin{aligned} S_1^-(q,\alpha ) e_L(\alpha ',q',\beta )= & {} q^{1/2} \alpha ^{-\frac{1}{2}(L-1)} \left[ q' \sigma ^-_1 \hat{n}_L - (\alpha ' \beta )^{-1} \hat{n}_1 \sigma ^-_L \right] q^{-S^z}\quad \end{aligned}$$

(134)

$$\begin{aligned} S_L^+(q,\alpha ) e_L(\alpha ',q',\beta )= & {} q^{1/2} \alpha ^{-\frac{1}{2}(L-1)} \left[ q' \hat{\upsilon }_1 \sigma ^+_L - (\alpha ' \beta )^{-1} \sigma ^+_1 \hat{\upsilon }_L \right] q^{S^z}\quad \end{aligned}$$

(135)

$$\begin{aligned} S_L^-(q,\alpha ) e_L(\alpha ',q',\beta )= & {} q^{-1/2} \alpha ^{\frac{1}{2}(L-1)} \left[ (q')^{-1} \hat{n}_1 \sigma ^-_L - \alpha ' \beta \sigma ^-_1 \hat{n}_L \right] q^{S^z}\quad \end{aligned}$$

(136)

and

$$\begin{aligned} e_L(\alpha ',q',\beta ) S_1^+(q,\alpha )= & {} q^{-1/2} \alpha ^{\frac{1}{2}(L-1)} \left[ q' \sigma ^+_1 \hat{n}_L - \alpha ' \beta \hat{n}_1 \sigma ^+_L \right] q^{-S^z}\end{aligned}$$

(137)

$$\begin{aligned} e_L(\alpha ',q',\beta ) S_1^-(q,\alpha )= & {} q^{1/2} \alpha ^{-\frac{1}{2}(L-1)} \left[ (q')^{-1} \sigma ^-_1 \hat{\upsilon }_L - (\alpha ' \beta )^{-1} \hat{\upsilon }_1 \sigma ^-_L \right] q^{-S^z}\quad \quad \quad \quad \end{aligned}$$

(138)

$$\begin{aligned} e_L(\alpha ',q',\beta ) S_L^+(q,\alpha )= & {} q^{1/2} \alpha ^{-\frac{1}{2}(L-1)} \left[ (q')^{-1} \hat{n}_1 \sigma ^+_L - (\alpha ' \beta )^{-1} \sigma ^+_1 \hat{n}_L \right] q^{S^z}\quad \end{aligned}$$

(139)

$$\begin{aligned} e_L(\alpha ',q',\beta ) S_L^-(q,\alpha )= & {} q^{-1/2} \alpha ^{\frac{1}{2}(L-1)} \left[ q' \hat{\upsilon }_1 \sigma ^-_L - \alpha ' \beta \sigma ^-_1 \hat{\upsilon }_L \right] q^{S^z}. \end{aligned}$$

(140)

Consider now $q=q'$ and $\alpha =\alpha '$. From the quantum algebra symmetry and from the previous relations one obtains (omitting the $q,\alpha$-dependence)

$$\begin{aligned} S^\pm H(\beta ) - H(\beta ') S^\pm= & {} S^\pm e_L(\beta ) - e_L(\beta ') S^\pm \\= & {} \left[ e_L(q^{-2}\beta ) - e_L(\beta ') \right] \sum _{k=2}^{L-1} S^\pm _k \nonumber \end{aligned}$$

(141)

$$\begin{aligned}&+ \left( S^\pm _1 + S^\pm _L \right) e_L(\beta ) - e_L(\beta ') \left( S^\pm _1 + S^\pm _L \right) . \end{aligned}$$

(142)

Observe that

$$\begin{aligned}&S_1^+ = q^{-1/2} \alpha ^{\frac{1}{2}(L-1)} \sigma ^+_1 q^{-S^z}, \quad S_1^- = q^{1/2} \alpha ^{-\frac{1}{2}(L-1)} \sigma ^-_1 q^{-S^z}, \end{aligned}$$

(143)

$$\begin{aligned}&S_L^+ = q^{1/2} \alpha ^{-\frac{1}{2}(L-1)} \sigma ^+_L q^{S^z}, \quad S_L^- = q^{-1/2} \alpha ^{\frac{1}{2}(L-1)} \sigma ^-_L q^{S^z} \end{aligned}$$

(144)

and the auxiliary relations

$$\begin{aligned} \sigma ^+_1 e_L(\beta )= & {} q^{-1} \sigma ^+_1 \hat{\upsilon }_L - \alpha \beta \hat{\upsilon }_1 \sigma ^+_L, \end{aligned}$$

(145)

$$\begin{aligned} e_L(\beta ) \sigma ^+_1= & {} q \sigma ^+_1 \hat{n}_L - \alpha \beta \hat{n}_1 \sigma ^+_L \end{aligned}$$

(146)

$$\begin{aligned} \sigma ^+_L e_L(\beta )= & {} q \hat{\upsilon }_1 \sigma ^+_L - (\alpha \beta )^{-1} \sigma ^+_1 \hat{\upsilon }_L, \end{aligned}$$

(147)

$$\begin{aligned} e_L(\beta ) \sigma ^+_L= & {} q^{-1} \hat{n}_1 \sigma ^+_L - (\alpha \beta )^{-1} \sigma ^+_1 \hat{n}_L, \end{aligned}$$

(148)

and

$$\begin{aligned} \sigma ^-_1 e_L(\beta )= & {} q \sigma ^-_1 \hat{n}_L - (\alpha \beta )^{-1} \hat{n}_1 \sigma ^-_L, \end{aligned}$$

(149)

$$\begin{aligned} e_L(\beta ) \sigma ^-_1= & {} q^{-1} \sigma ^-_1 \hat{\upsilon }_L - (\alpha \beta )^{-1} \hat{\upsilon }_1 \sigma ^-_L \end{aligned}$$

(150)

$$\begin{aligned} \sigma ^-_L e_L(\beta )= & {} q^{-1} \hat{n}_1 \sigma ^-_L - \alpha \beta \sigma ^-_1 \hat{n}_L, \end{aligned}$$

(151)

$$\begin{aligned} e_L(\beta ) \sigma ^-_L= & {} q \hat{\upsilon }_1 \sigma ^-_L - \alpha \beta \sigma ^-_1 \hat{\upsilon }_L. \end{aligned}$$

(152)

Thus one obtains

$$\begin{aligned}&\left( S^+_1 + S^+_L \right) e_L(\beta ) - e_L(\beta ') \left( S^+_1 + S^+_L \right) \nonumber \\&= A^+(\beta ,\beta ') \beta ^{1/2} \alpha ^{L/2} q^{-S^z-1} + B^+(\beta ,\beta ') \beta ^{-1/2} \alpha ^{-L/2} q^{S^z+1} \end{aligned}$$

(153)

with

$$\begin{aligned} A^+(\beta ,\beta ')= & {} \frac{q^{1/2}}{(\alpha \beta )^{1/2}}\sigma ^+_1 \left[ q^{-1} \hat{\upsilon }_L - q \hat{n}_L\right] - \frac{(\alpha \beta )^{1/2}}{q^{1/2}} \sigma ^+_L \left[ q \hat{\upsilon }_1 - q \frac{\beta '}{\beta }\hat{n}_1\right] \end{aligned}$$

(154)

$$\begin{aligned} B^+(\beta ,\beta ')= & {} \frac{q^{1/2}}{(\alpha \beta )^{1/2}}\sigma ^+_1 \left[ q^{-1} \frac{\beta }{\beta '}\hat{n}_L - q^{-1} \hat{\upsilon }_L\right] - \frac{(\alpha \beta )^{1/2}}{q^{1/2}} \sigma ^+_L \left[ q^{-1} \hat{n}_1 - q \hat{\upsilon }_1\right] \quad \quad \quad \quad \quad \end{aligned}$$

(155)

and

$$\begin{aligned}&\left( S^-_1 + S^-_L \right) e_L(\beta ) - e_L(\beta ') \left( S^-_1 + S^-_L \right) \nonumber \\&= A^-(\beta ,\beta ') \beta ^{-1/2} \alpha ^{-L/2} q^{-S^z+1} + B^-(\beta ,\beta ') \beta ^{1/2} \alpha ^{L/2} q^{S^z-1} \end{aligned}$$

(156)

with

$$\begin{aligned} A^-(\beta ,\beta ')= & {} \frac{(\alpha \beta )^{1/2}}{q^{1/2}}\sigma ^-_1 \left[ q \hat{n}_L - q^{-1} \hat{\upsilon }_L\right] - \frac{q^{1/2}}{(\alpha \beta )^{1/2}} \sigma ^-_L \left[ q^{-1} \hat{n}_1 - q^{-1} \frac{\beta }{\beta '}\hat{\upsilon }_1\right] \quad \quad \quad \quad \quad \end{aligned}$$

(157)

$$\begin{aligned} B^-(\beta ,\beta ')= & {} \frac{(\alpha \beta )^{1/2}}{q^{1/2}}\sigma ^-_1 \left[ q \frac{\beta '}{\beta }\hat{\upsilon }_L - q \hat{n}_L\right] - \frac{q^{1/2}}{(\alpha \beta )^{1/2}} \sigma ^-_L \left[ q \hat{\upsilon }_1 - q^{-1} \hat{n}_1\right] . \end{aligned}$$

(158)

With the choice $\beta ' = q^{-2} \beta$ (142) reduces to

$$\begin{aligned} S^\pm H(\beta ) - H(q^{-2} \beta ) S^\pm = \left( S^\pm _1 + S^\pm _L \right) e_L(\beta ) - e_L(q^{-2}\beta ) \left( S^\pm _1 + S^\pm _L \right) . \end{aligned}$$

(159)

For $S^+$ the r.h.s. reduces to

$$\begin{aligned}&\left\{ \frac{q^{1/2}}{(\alpha \beta )^{1/2}}\sigma ^+_1 \left[ q^{-1} \hat{\upsilon }_L - q \hat{n}_L\right] - \frac{(\alpha \beta )^{1/2}}{q^{1/2}} \sigma ^+_L \left[ q \hat{\upsilon }_1 - q^{-1} \hat{n}_1\right] \right\} \nonumber \\&\quad \times \left[ \beta ^{1/2} \alpha ^{L/2} q^{-S^z-1} - \beta ^{-1/2} \alpha ^{-L/2} q^{S^z+1} \right] \nonumber \end{aligned}$$

With (143), (144) one thus arrives at

$$\begin{aligned} S^+ H(\beta ) - H(q^{-2} \beta ) S^+= & {} \left[ q^{-1} \hat{\upsilon }_L - q \hat{n}_L\right] S^+_1 \left[ 1-\beta ^{-1}\alpha ^{-L} q^{2S^z+2} \right] \quad \nonumber \\&+ \left[ q \hat{\upsilon }_1 - q^{-1} \hat{n}_1\right] S^+_L \left[ 1-\beta \alpha ^{L} q^{-2S^z-2} \right] . \end{aligned}$$

(160)

Notice that the action of the pseudo commutator on states with particle number satisfying

$$\begin{aligned} q^{L-2N+2} = \beta \alpha ^L \end{aligned}$$

(161)

vanishes.

Similarly one obtains for $S^-$ the r.h.s. of (159)

$$\begin{aligned}&\left\{ \frac{(\alpha \beta )^{1/2}}{q^{1/2}}\sigma ^-_1 \left[ q^{-1} \hat{\upsilon }_L - q \hat{n}_L\right] - \frac{q^{1/2}}{(\alpha \beta )^{1/2}} \sigma ^-_L \left[ q \hat{\upsilon }_1 - q^{-1} \hat{n}_1\right] \right\} \nonumber \\&\quad \times \left[ \beta ^{1/2} \alpha ^{L/2} q^{S^z-1} - \beta ^{-1/2} \alpha ^{-L/2} q^{-S^z+1} \right] \nonumber \end{aligned}$$

which yields

$$\begin{aligned} S^- H(\beta ) - H(q^{-2} \beta ) S^-= & {} -\left[ q^{-1} \hat{\upsilon }_L - q \hat{n}_L\right] S^-_1 \left[ 1-\beta \alpha ^{L} q^{2S^z-2} \right] \nonumber \\&- \left[ q \hat{\upsilon }_1 - q^{-1} \hat{n}_1\right] S^-_L \left[ 1-\beta ^{-1} \alpha ^{-L} q^{-2S^z+2} \right] .\quad \quad \quad \end{aligned}$$

(162)

Notice that the action of the pseudo commutator on states with particle number satisfying

$$\begin{aligned} q^{-L+2N+2} = \beta \alpha ^L \end{aligned}$$

(163)

vanishes.

In compact form (159) can thus be written

$$\begin{aligned} S^\pm H(\beta ) - H(q^{-2} \beta ) S^\pm= & {} \pm \left[ q^{-1} \hat{\upsilon }_L - q \hat{n}_L\right] S^\pm _1 \left[ 1-\beta ^{\mp 1}\alpha ^{\mp L} q^{2S^z\pm 2} \right] \nonumber \\&\pm \left[ q \hat{\upsilon }_1 - q^{-1} \hat{n}_1\right] S^\pm _L \left[ 1-\beta ^{\pm 1} \alpha ^{\pm L} q^{-2S^z\mp 2} \right] .\quad \quad \quad \end{aligned}$$

(164)

One can iterate. E.g. for $(S^-)^2$ one obtains

$$\begin{aligned}&(S^-)^2 H(\beta ) - H(q^{-4} \beta ) (S^-)^2 \nonumber \\&\quad = (1+q^{-2}) \left[ q \hat{n}_L - q^{-1} \hat{\upsilon }_L\right] S^-_1 \left( \sum _{k=2}^{L-1} S^-_k\right) \left[ 1-\beta \alpha ^{ L} q^{2S^z -4} \right] \nonumber \\&\quad + (1+q^{-2})\left[ q^{-1} \hat{n}_1 - q \hat{\upsilon }_1\right] \left( \sum _{k=2}^{L-1} S^-_k\right) S^-_L \left[ 1-\beta ^{- 1} \alpha ^{- L} q^{-2S^z+4} \right] .\quad \end{aligned}$$

(165)

Iterating further as in [30] one arrives at Proposition 1.