Similarity revisited: shock random walks in the asymmetric simple exclusion process with open boundaries

Abstract

A reverse duality between the asymmetric simple exclusion process (ASEP) with open boundary conditions and a biased random walk of a single particle is proved for a special manifold of boundary parameters of the ASEP. The duality function is given by the configuration probabilities of a family of Bernoulli shock measures with a microscopic shock at site x of the lattice. The boundary conditions of the dual random walk, which can be reflecting or absorbing, depend on the choice of the duality function. As a consequence of this duality, the full time-dependent distribution of the open ASEP starting from a fixed shock measure at x is given for any time $t>0$ by a convex combination of shock measures with shock at position y. The coefficients are the transition probabilities P(x, t|y, 0) at time t of the random walk starting at site y. It is also shown that this family of shock measures arises from a similarity transformation of the two-dimensional representation of the matrix algebra for the stationary matrix product measure of the open ASEP. This implies a further duality between random walks with different boundary conditions.

1 Introduction

This paper is devoted to Malte Henkel on the occasion of his 60th birthday. The main themes are duality and the microscopic structure and dynamics of macroscopic shock waves in driven diffusive systems. Duality is closely related to the notion of similarity [1] between Markovian stochastic lattice gas models, for which we adopt the generic term “stochastic interacting particle systems” commonly used in the probabilistic literature devoted to the subject [2]. The investigation of shocks is part of the long-standing quest for understanding in a nonequilibrium setting the nature of macroscopic discontinuities on microscopic scale [3]. Duality has turned out to be a powerful tool for tackling this problem and is intricately linked to symmetries of integrable quantum spin systems closely related to stochastic interacting particle systems [4,5,6,7].

Indeed, Markov processes with countable state space are often defined by the intensity matrix M whose off-diagonal matrix elements are the transition rates from one state to another and whose diagonal elements are the negative sum of all outgoing rates so that the sum of all elements in a row of M add up to zero. These two properties express positivity of the transition rates and conservation of probability. Alternatively, the Markov generator can be written as the negative transpose $H:=-M^T$ of the intensity matrix, and for many Markov processes of interest, the matrix H happens to be very closely related to (or even identical with) the quantum Hamiltonian operator of a quantum system. This fortuitous coincidence, perhaps first pointed out by Glauber [8], can often be used to apply techniques from quantum mechanics to treat purely classical Markovian dynamics. This is explained and applied in a vast body of literature; see, e.g., [4,5,6,7] and references therein. The present work deals exclusively with classical stochastic interacting particle systems for which the quantum Hamiltonian formalism has turned out to be particularly fruitful. Hence, to be consistent with this literature, we use the quantum Hamiltonian representation $H=-M^T$ of the Markov generator M.

The current mathematical feast is structured as follows: The starter (Sect. 2) reviews in a pedestrian way the concept of duality and how it is related to similarity between Markov processes and also discusses the significance of shocks in the framework of the asymmetric simple exclusion process. The first course, presented in Sect. 3, defines Bernoulli shock measures, discusses some of their properties and shows how they arise from a family of two-dimensional representations of the matrix algebra for the open ASEP. As the second course, we derive the main result of this paper, i.e., a reverse duality w.r.t. to duality functions given by the shock measures, and discuss some consequences of this duality (Sect. 4). Some conclusions and comments on open questions are presented as cheese and desert in Sect. 5. The dishes are spiced with a sprinkle of quantum spin chain physics and accompanied by an extensive list of selected references that includes some mathematical background literature.

2 Duality and shock random walks: a brief review

2.1 Similarity and duality

It is well known that many generators of stochastic interacting particle systems are related by a similarity transformation $W = A^{-1} Q A$ between the intensity matrices W of one process and Q of the other [1]. If at least one of the processes is reversible then similarity is a special case of duality between interacting particle systems [2], which has become a topic of intense research during the last decade; see, e.g., [9,10,11,12,13,14,15,16,17] and references therein for various applications to driven diffusive systems and reaction–diffusion processes, and specifically [18,19,20,21,22,23,24,25] for very recent work on duality for models closely related to the ASEP that is the topic of the present work.

To see the link between similarity and duality and to make contact with [1], we denote by $H := - W^T$ the quantum Hamiltonian representation [4, 5, 7] of the intensity matrix W and similarly $G := - Q^T$ for a generally different process with intensity matrix Q. With $B:=A^T$, the similarity transformation reads

$$\begin{aligned} H = B G B^{-1}. \end{aligned}$$

(2.1)

On the other hand, duality for two Markov processes $\varvec{\eta }(t)$ and $\textbf{x}(t)$ with countable state spaces is expressed by the intertwining relation

$$\begin{aligned} D H = G^T D, \end{aligned}$$

(2.2)

where the duality matrix D has the duality function $D(\textbf{x},\varvec{\eta }) = D_{\textbf{x}\varvec{\eta }}$ as matrix elements [26], where the indices $\textbf{x}$ and $\varvec{\eta }$ can be represented by numbers i, j ranging from 1 to d and $d'$, respectively, where d and $d'$ are the cardinalities of the respective state spaces. Assume now that the Markov process $\varvec{\eta }(t)$ of states $\varvec{\eta }$ is generated by the quantum Hamiltonian H and satisfies detailed balance w.r.t. a stationary distribution $\mu ^*_{\varvec{\eta }} > 0$ for all $\varvec{\eta }$. With the diagonal matrix $\hat{\mu }^*$ that has the stationary weights $\mu ^*_{\varvec{\eta }}$ on the diagonal, the time-reversed process is given by definition by the generator $H^{rev} = \hat{\mu }^*H^T \hat{\mu }^{*-1}$. Then, the similarity (2.1) with transformation matrix B to another process $\varvec{\xi }(t)$ generated by a quantum Hamiltonian G turns into the duality relation $D H^{rev} = G^T D$ between the time-reversed process generated by $H^{rev}$ and the process generated by $G =B^{-1} H B$ with duality matrix $D = B \hat{\mu }^{*-1}$. If the process $\varvec{\eta }(t)$ is reversible, i.e., if $H^{rev} = H$, then one obtains from similarity directly the duality (2.2).

Likewise, let us assume that the process $\varvec{\xi }(t)$ generated by G satisfies detailed balance with stationary distribution $\pi ^*_{\varvec{\xi }} > 0$ for all $\varvec{\xi }$. With the diagonal matrix $\hat{\pi }^*$ with the stationary weights $\pi ^*_{\varvec{\xi }}$ on the diagonal, similarity yields $H R = R (G^{rev})^T$ with a duality matrix $R = B \hat{\pi }^*$. For reversible $\varvec{\xi }(t)$, i.e., $G^{rev} = G$, similarity thus turns into

$$\begin{aligned} H R = R G^T. \end{aligned}$$

(2.3)

Since this intertwining relation is related to duality for the time-reversed processes, we call it reverse duality [25].

Duality or reverse duality, however, is stronger than similarity since it requires the existence of neither the inverse transformation matrix B nor an invertible stationary distribution for one of the two processes. In fact, the duality matrix does not even need to be a square matrix. Thus, duality and reverse duality can be used to relate processes with different state space. Analogously, one can extend the notion of similarity to the intertwining relation

$$\begin{aligned} H S = S G, \end{aligned}$$

(2.4)

which for invertible S reduces to similarity with $B = S$ and to a symmetry expressed by the commutation relation $[H,S]=0$ for the generator of a self-dual process where $G=H$ [27,28,29,30]. We conclude that similarity, duality, symmetry, and reverse duality are closely related notions.

Similarity allows for relating classical Markov processes to integrable quantum spin systems [6] and thus makes available the full machinery of integrability in the Yang–Baxter sense [31,32,33] for the study of interacting particle systems. Duality then makes it possible to express time-dependent expectations for one Markov process in terms of time-dependent expectations of another Markov process and is thus an important tool for the study of stochastic interacting particle systems, particularly if the dual process has some simple properties that are “hidden” in the original process.

To see this, consider a Markov process $\varvec{\eta }(t)$ generated by H, a dual Markov process $\textbf{x}(t)$ generated by G and consider the duality function to represent an indexed family of functions $f^{\textbf{x}}(\varvec{\eta }):=D(\textbf{x},\varvec{\eta })$ of the states $\varvec{\eta }$. Denoting the expectation of such a function, indexed by $\textbf{x}$, at time $t\ge 0$ w.r.t. some unspecified initial distribution by ${\langle \, {f^{\textbf{x}}(t)}\, \rangle }$ and iterating (2.2) yields $D \textrm{e}^{-Ht}= \textrm{e}^{-G^T t} D$ for the duality matrix D. Therefore, one obtains

$$\begin{aligned} {\langle \, {f^{\textbf{x}}(t)}\, \rangle } = \sum _{\textbf{y}} P(\textbf{x},t|\textbf{y},0) {\langle \, {f^{\textbf{y}}(0)}\, \rangle }. \end{aligned}$$

(2.5)

with the transition probability $P(\textbf{x},t | \textbf{y},0) = {\langle \, {\textbf{x}}\, |} \textrm{e}^{-Gt} {| \, {\textbf{y}}\, \rangle }$ for a transition from state $\textbf{y}$ at time $t=0$ to a state $\textbf{x}$ at time $t\ge 0$ of the dual process. Thus, given a duality function, duality allows for computing for arbitrary initial distributions $\mu _{\varvec{\eta }}$, the time-dependent expectations of certain functions $f^{\textbf{x}}(\varvec{\eta })$, encoded in the duality function, in terms of transition probabilities of the dual process.

A different but also powerful property originates in the reverse duality relation (2.3) and in the intertwining relation (2.4), where the duality matrix appears to the right of the generator H. Iterating (2.4) yields $\textrm{e}^{-Ht} S = S \textrm{e}^{-G t}$ for the Markovian time evolution of the two processes. Hence, for the probability $\mu ^{\textbf{x}}_{\varvec{\eta }}(t) = \sum _{\varvec{\xi }} S_{\varvec{\xi }\textbf{x}} {\langle \, {\varvec{\eta }}\, |} \textrm{e}^{-Ht} {| \, {\varvec{\xi }}\, \rangle }$ of finding the state $\varvec{\eta }$ at time t for a family of initial distributions $\mu ^{\textbf{x}}_{\varvec{\eta }} = S_{\varvec{\eta }\textbf{x}}$, one finds

$$\begin{aligned} \mu ^{\textbf{x}}_{\varvec{\eta }}(t) = \sum _{\textbf{y}} \mu ^{\textbf{y}}_{\varvec{\eta }} P(\textbf{y},t | \textbf{x},0). \end{aligned}$$

(2.6)

with the transition probability $P(\textbf{y},t | \textbf{x},0)$ of the dual process.

In a similar fashion, one obtains from reverse duality (2.3) with the duality function $R_{\varvec{\eta }\textbf{x}}=\mu ^{\textbf{x}}_{\varvec{\eta }}$ the time evolution property

$$\begin{aligned} \mu ^{\textbf{x}}_{\varvec{\eta }}(t) = \sum _{\textbf{y}} \mu ^{\textbf{y}}_{\varvec{\eta }} P(\textbf{x},t | \textbf{y},0). \end{aligned}$$

(2.7)

In other words, given a duality function, the intertwining relation (2.4) and similarly reverse duality (2.3) allow for computing the time-dependent expectation of arbitrary functions $f(\varvec{\eta })$ for certain initial distributions $\mu ^{\textbf{x}}$ in terms of the transition probability of the dual process.

We stress, however, that taking $S_{\varvec{\eta }\textbf{x}}$ or $R_{\varvec{\eta }\textbf{x}}$ to be a measure, i.e., a normalized and strictly nonnegative family of functions of the configurations $\varvec{\eta }$, is just one example of the use of intertwining and reverse duality, whereas the time evolution formulas (2.6) and (2.7) are valid for any family of duality functions.

2.2 The setting: the open asymmetric simple exclusion process

Here, we exploit intertwining duality for the one-dimensional open partially asymmetric simple exclusion process (ASEP) [34,35,36,37,38,39] with L sites labeled from 1 to L. Particles jump randomly with rate $r>0$ to the right neighboring site and with rate $\ell >0$ to the left neighboring site, provided the target site is empty, thus respecting the exclusion rule of at most one particle per site. At the boundaries, the particle number is not conserved. Particles are injected with rate $\alpha$ ($\delta$) on the left (right) boundary site 1 (L) and absorbed with rate $\gamma$ ($\beta$) on the left (right) boundary site, again respecting the exclusion rule.

For this process, we prove below some intertwining dualities with a simple biased one-dimensional random walk with reflecting boundaries or absorbing boundaries. In either case, analysis of the random walk problem is basic textbook material and one obtains by simple means highly nontrivial information about the full time evolution of the ASEP with open boundaries.

To be definite, we assume without loss of generality a hopping bias to the right (i.e., $r>\ell$) and define the jump asymmetry

$$\begin{aligned} q := \sqrt{\frac{r}{\ell }}, \end{aligned}$$

(2.8)

and the functions

$$\begin{aligned} \kappa _\pm (x,y) := \frac{y-x+r-\ell \pm \sqrt{(y-x+r-\ell )^2 + 4xy}}{2x}, \end{aligned}$$

(2.9)

which are convenient for the characterization of the stationary distribution [39]. We consider the parameter manifold [40]

$$\begin{aligned} \kappa _+(\alpha ,\gamma ) \kappa _+(\beta ,\delta ) = q^{2N}, \end{aligned}$$

(2.10)

with $N=1$ and a family of inhomogeneous initial distributions that are uncorrelated but have a density discontinuity at a lattice site x with a jump from density $\rho _0$ for sites $k<x$ to density $\rho _1>\rho _0$ for sites $k>x$ and a special state $\rho ^\star$ at site x that we refer to as a shock position. Following [41], we refer to such a distribution, defined precisely below, as a Bernoulli shock measure (BSM) with shock at site x and denote it by $\mu ^x_{\varvec{\eta }}(\rho ^\star )$. Using reverse duality and intertwining duality, it turns out that, in the open ASEP, these shock measures maintain their internal structure. Only the shock position changes by evolving with a simple random walk dynamics specified below.

2.3 The phenomenon: shock random walk

The appearance of a shock random walk in the open ASEP is not entirely surprising. It was been conjectured in the context of the exact solution of the stationary distribution of the open totally asymmetric simple exclusion process (TASEP) [38] to explain the appearance of the first-order nonequilibrium phase transition between low-density phase and high-density phase. Later, this picture was used in [42] to explain properties of current fluctuations in the TASEP, and in [43] it was used to elucidate the experimentally observed slowing down of ribosomes during the process of protein synthesis for which the ASEP was initially introduced as a simple model [44]. Indeed, the notion of a shock random walk has become a popular paradigm for fluctuating traffic jams in complex systems [45,46,47,48,49,50,51].

The picture of a random walking shock was subsequently probed more quantitatively by investigating the spectral gaps of the generator H [52,53,54], which in certain regions of parameter space indeed yields eigenvalues consistent with what one expects from biased random walk dynamics. In [55], a shock random walk dynamics was then directly derived by placing a second-class particle [56] at the shock position and extending the lattice of the open ASEP by reservoir sites at each boundary with the boundary rates on a one-parameter submanifold $\mathcal {A}_1 \subset \mathcal {B}_1$ defined by the choice $\alpha = r \rho _0$, $\gamma = \ell (1-\rho _0)$, $\delta = \ell \rho _1$, $\gamma = r(1-\rho _1)$ with

$$\begin{aligned} \frac{\rho _1(1-\rho _0)}{\rho _0(1-\rho _1)} = q^2, \end{aligned}$$

(2.11)

for the boundary parameters $\rho _{0,1}$, which can be understood as reservoir densities to which the ASEP is connected at its boundaries.

Instead of placing a second-class particle on the shock position, one may also take the shock state $\rho ^\star$ to be a state with density $\rho _0$ [41]. For a finite system, this implies that a shock at site L corresponds to a homogeneous Bernoulli product measure with constant density $\rho _0$ [55]. Also an extra “shock” state with no shock given by a homogeneous Bernoulli product measure with density $\rho _1$ was introduced and defined to represent a shock at an auxiliary site $x=0$. In [57], it was then shown that Bernoulli shock measures $\mu ^{\textbf{x}}$ with N shocks at a set of positions $\textbf{x}$ with $0 \le x_1< x_2< \dots < x_N \le L$ appear in the stationary matrix product measures (MPM) of the open ASEP on the manifold (2.10), where they can be expressed in terms of $(N+1)$-dimensional representations [40] of the matrix algebra developed for an alternative exact solution of the stationary distribution of the open ASEP [37]. Specifically, the two-dimensional representation of [40], discussed with a different representation already in [58], is a convex combination of Bernoulli shock measures with a single shock. As explanation for this observation, a shock random walk dynamics with reflecting boundaries was proposed in [57], but neither was this proved nor were the postulated boundary jump rates of the shock specified.

In [59], this approach was extended to a totally asymmetric simple exclusion process with deterministic discrete-time sublattice update and stochastic injection and absorption at the boundaries [60]. For this model, a discrete-time random walk dynamics for shock measures was rigorously derived and the stationary distribution of the random walk was shown to be given by the two-dimensional representation of the stationary matrix algebra derived in [61], which thus can also be expressed as a convex combination of shock measures with a single shock.

In [25], the open ASEP is studied on a submanifold defined by the more general boundary constraint (2.10) for $(N+1)$-dimensional representations for the MPM together with the further constraint

$$\begin{aligned} \kappa _-(\alpha ,\gamma ) \kappa _-(\beta ,\delta ) = q^{-2k} \end{aligned}$$

(2.12)

on the boundary rates [40, 58, 62]. For $k=1$, which is is equivalent to

$$\begin{aligned} \alpha \beta =\gamma \delta , \end{aligned}$$

(2.13)

a reverse duality is proved rigorously for N consecutive shocks with a specific choice of bulk densities $\rho _i$ between shocks and shock densities $\rho ^\star _i$ (which all have to satisfy the analog of (2.11) for subsequent indices $i-1$ and i). This duality asserts that the dual process is an interacting random walk of N particles on a lattice of L sites with conservative reflecting boundaries. Thus, from a qualitative viewpoint, the picture is similar to what one may expect from putting second-class particles at the shock position or choosing shock densities $\rho ^\star _i=\rho _i$. However, no auxiliary sites need to be introduced.

Shock random walks on the infinite lattice are known to appear not only in the ASEP [41] and its n-species generalization with lower-class particles [18, 63], but also in other conservative interacting particle systems [64, 65]. From a physics perspective, this is an interesting phenomenon not only as a model for traffic jams but also for another reason: The stability of the shock on microscopic scale during the time evolution elucidates the fine structure of the macroscopic shock discontinuity that appears in the large-scale hydrodynamic description [66].

Indeed, for the ASEP, one can show that, even though the microscopic shock random walk requires the fine-tuning (2.11) of the two shock densities, the lack of correlations in the evolving probability distribution can be proved generally at large lattice distances by looking at the density distribution as seen from a second-class particle that moves with the speed of a shock and thus can be used a microscopic definition of the shock position [56, 67]. Also, the shock velocity and shock diffusion coefficient obtained from the microscopic random walk dynamics [41] for fine-tuned shock densities remain valid for general shock densities [68]. Only the macroscopic Rankine–Huginot condition—which for the ASEP only requires $\rho _1 > \rho _0$ for $q>1$—needs to be satisfied for the shock to be microscopically stable [69].

We also note that a shock can be seen as a nonequilibrium analog of a domain wall in phase-separated states, which have been studied in a variety of driven lattice gas models; see, e.g., [70,71,72,73,74]. In contrast to the phase-separated states in these nonequilibrium systems where the domain wall is microscopically sharp but immobile and connects frozen domains, a shock (like in the ASEP) connects fluctuating domains and its position may fluctuate. Moreover, a nonequilibrium domain wall between states with densities $\rho _{0,1}$ is stable both for $\rho _1 > \rho _0$ and for $\rho _1 < \rho _0$, while a shock is generally stable for only one of these conditions as the other would then automatically violate the Rankine–Huginot condition and evolve into a rarefaction fan whose width increases proportionally to time and inside which the location of a second-class particle shock marker is distributed uniformly [68].

2.4 Synopsis

The aim of the present work is to fill some of the gaps that the literature surveyed above has left in the study of shock random walks in the open ASEP. First we note that, in [25], a family of two-dimensional submanifolds in the four-dimensional space of boundary parameters is studied, which is defined by the relations (2.13) and (2.10) for $N=1,\dots ,L$. This yields the time evolution of shock measures with N shocks. In contrast, here we consider the three-dimensional submanifold in the four-dimensional space of boundary parameters defined by only the condition (2.10) with $N=1$. Focusing on a single shock, we obtain the following main results:

• 1. We present in detail how similarity transformations of the two-dimensional representation of the matrix algebra for the open ASEP obtained in [40] can be written in explicit form as a convex combination of Bernoulli shock measures with any choice of shock state. Thus we go beyond [57], where only a shock state with shock density $\rho ^\star =\rho _0$ is considered. For the special submanifold (2.13) of (2.10) for $N=1$, we go beyond [25, 62], where the link between the MPM and Bernoulli shock measures is not studied.

• 2. We derive random walk dynamics with reflecting boundaries for generic shock densities in the range $\rho _0<\rho ^\star <\rho _1$ and compute the boundary rates, thus going beyond [57], where the random walk dynamics with reflecting boundaries is only postulated but not derived. We also go beyond [25], where only the submanifold (2.13) and $\rho ^\star =\alpha /(\alpha +\gamma )$ is considered.

• 3. We obtain a new intertwining duality with absorbing boundaries reminiscent of the absorbing dual of the symmetric simple exclusion process with open boundaries [12, 22, 75]. This duality involves a difference of two shock measures in the construction of the duality function and can be used to compute further eigenvectors of the generator. From point 1, the nature of the mathematically precise but physically rather obscure MPM is elucidated: A typical stationary configuration as described by a two-dimensional representation of the MPM consists of a region of k lattice sites with low particle density $\rho _0 < 1/2$ (beginning at the left boundary) and is followed by a region of high particle density $\rho _1 > 1/2$ (ending at the right boundary). The transition from the low-density domain to the high-density domain is microscopically sharp, i.e., extends over a few lattice sites, which is completely analogous to a domain wall in spin systems, as noted above. The stationary position k of the domain wall is random and geometrically distributed over the lattice. The set of all such states is the convex combination of Bernoulli shock measures with shock position k described by the MPM. Moreover, we learn that the value $\rho ^\star$ of the shock density does not have any physical significance. It only means that the microscopic shock position k can be defined by states with no particle at site k, or by states with exactly one particle at site k, or by a linear combination of two such states. From point 2, we learn that the random shock position performs a lattice random walk with generically reflecting boundary conditions such that the stationary distribution of this random walk is the geometric distribution of the BSMs with shock at site k. Finally, point 3 shows that, for a special linear combination of BSMs, the random walk dynamics of the shock position can be expressed by absorbing boundary conditions. This at first sight surprising result is analogous to the well-known equivalence of absorbing and reflecting barrier problems for random walks known as Siegmund duality [76]. In this case, the absorbing state represents the stationary geometric distribution of the shock position.

3 Matrix product state and shock measures

In this section, we introduce the Bernoulli shock measures and derive some of their physical properties, in particular the density profile and current (Sect. 3.1). Then, we go on to explain in Sect. 3.2 that the stationary distribution of the ASEP given by two-dimensional matrix representation obtained in [40] for the matrix algebra of [37] is a convex combination of Bernoulli shock measures. In particular, we show that a change of basis in the representation matrices changes the shock density in the Bernoulli shock measure.

Throughout this work, we apply for numbers $a_k \in {\mathbb C}$ indexed by $k\in {\mathbb Z}$ the empty-sum convention $\sum _{k=n+1}^{n} a_k = 0$ for any $n\in {\mathbb Z}$, which for $b_k = \exp {(a_k)}$ implies the empty-product convention $\prod _{k=n+1}^{n} b_k = 1$ for any $n\in {\mathbb Z}$.

We denote a state of the ASEP by the occupation numbers $\eta _k\in \{0,1\}$ and the whole configuration of particles on the lattice by $\varvec{\eta }:= (\eta _1,\eta _2, \dots ,\eta _L)$. For the boundary rates, we choose the parameterization [25]

$$\begin{aligned} \begin{array}{lcl} \alpha &{} = &{} (r+\omega _-)\rho _- \\ \gamma &{} = &{} (\ell +\omega _-)(1-\rho _-) \\ \beta &{} = &{} (r+\omega _+)(1-\rho _+) \\ \delta &{} = &{} (\ell +\omega _+) \rho _+, \end{array} \end{aligned}$$

(3.1)

with boundary densities $\rho _{\pm }$ and the quantities $\omega _{\pm }$, which can be interpreted as boundary barriers since $\omega _{\pm }=0$ would corresponds to biased hopping with the same rates $r,\ell$ as in the bulk, but from reservoirs of deterministic densities $\rho _\pm$. In terms of fugacities

$$\begin{aligned} z_{\pm } := \frac{\rho _{\pm }}{1-\rho _{\pm }}, \end{aligned}$$

(3.2)

one has

$$\begin{aligned} \kappa _+(\alpha ,\gamma ) = z_-^{-1}, \quad \kappa _+(\beta ,\delta ) = z_+ , \end{aligned}$$

(3.3)

independently of $\omega _{\pm }$ and

$$\begin{aligned} \kappa _-(\alpha ,\gamma ) = - \frac{\ell +\omega _-}{r+\omega _-}, \quad \kappa _-(\beta ,\delta ) = - \frac{\ell +\omega _+}{r+\omega _+}, \end{aligned}$$

(3.4)

independently of $z_{\pm }$.

3.1 Shock measures

With the marginals

$$\begin{aligned} B_k(\rho ) := (1-\rho )(1-\eta _k) + \rho \eta _k, \end{aligned}$$

(3.5)

the shock measures with shock at site x and shock density $\rho ^\star$ as described in the “Introduction” are given by

$$\begin{aligned} \mu ^{x}_{\varvec{\eta }}(\rho ^\star ) = \left\{ \begin{array}{ll} \displaystyle \prod _{k=1}^{L} B_k(\rho _1) &{} x=0 \\ \displaystyle \prod _{k=1}^{x-1} B_k(\rho _0) B_x(\rho ^\star ) \prod _{k=x+1}^{L} B_k(\rho _1) &{} 1\le x \le L \\ \displaystyle \prod _{k=1}^{L} B_k(\rho _0) &{} x=L+1 \end{array} \right. . \end{aligned}$$

(3.6)

With the function

$$\begin{aligned} N_k(\varvec{\eta }) := \sum _{l=1}^k \eta _k, \quad N(\varvec{\eta }) := N_L(\varvec{\eta }), \end{aligned}$$

(3.7)

and the fugacities $z_{0,1}$ defined as in (3.2) as functions of the densities $\rho _{0,1}$, one can write

$$\begin{aligned} \mu ^{x}_{\varvec{\eta }}(\rho ^\star ) = A_L(\rho ^\star ) \left( \frac{1-\rho _0}{1-\rho _1}\right) ^{x-1} z_1^{N(\varvec{\eta })} \left( \frac{z_0}{z_1}\right) ^{N_{x-1}(\varvec{\eta })} B_x(\tau ^\star ), \end{aligned}$$

(3.8)

with the constants

$$\begin{aligned} A_L(\rho ^\star ) = \frac{(1-\rho ^\star )\rho _1 + (1-\rho _1)\rho ^\star }{\rho _1(1-\rho _1)}(1-\rho _1)^{L} , \quad \tau ^\star = \frac{z^\star }{z^\star + z_1}. \end{aligned}$$

(3.9)

This representation of the BSM follows from the trivial identity $z^\eta = 1-\eta + z \eta$ for $\eta \in \{0,1\}$. In the following, we point out further properties of these shock measures.

3.1.1 Linear dependence

Of these $L+2$ shock measures, only $L+1$ are linearly independent. This can be seen immediately by taking $\rho ^\star = \rho _0$. Then by construction $\mu _{\rho _0}^{L+1}=\mu _{\rho _0}^{L}$, while the linear independence of the remaining $\mu ^{x}(\rho _0)$ follows from the product form of linearly independent marginals.

To understand the linear dependence in the general case, we note that from the definition (3.5), one obtains for $1 \le x \le L +1$

$$\begin{aligned} \mu ^{x}(\rho _1) = \mu ^{x-1}(\rho _0), \quad \mu ^{x}(\rho ^\star ) = \chi \mu ^{x}(\rho _1) + (1-\chi ) \mu ^{x}(\rho _0), \end{aligned}$$

(3.10)

with the coefficient

$$\begin{aligned} \chi = \frac{\rho ^\star -\rho _0}{\rho _1-\rho _0}. \end{aligned}$$

(3.11)

With an arbitrary coefficient $\psi$, this yields for any $\rho ^\star$ the decomposition of the $L+2$ shock measures

$$\begin{aligned} \mu ^{0}(\rho ^\star ) & = \mu ^{0}(\rho _0), \end{aligned}$$

(3.12)

$$\begin{aligned} \mu ^{x}(\rho ^\star ) & = \chi \mu ^{x-1}(\rho _0) + (1-\chi ) \mu ^{x}(\rho _0), \quad 1 \le x \le L, \end{aligned}$$

(3.13)

$$\begin{aligned} \mu ^{L+1}(\rho ^\star ) & = \psi \mu ^{L}(\rho _0) + (1-\psi ) \mu ^{L+1}(\rho _0) \, = \, \mu ^{L}(\rho _0), \end{aligned}$$

(3.14)

in terms of the $L+1$ linearly independent shock measures $\mu ^{x}(\rho _0)$. This is a convex combination for $\rho _0 \le \rho ^\star \le \rho _1$. In matrix form with the $(L+2)$-dimensional matrix M with matrix elements

$$\begin{aligned} M_{yx} & = \left[ 1-\chi + \chi \delta _{x,0} + (\chi -\psi ) \delta _{x,L+1} \right] \delta _{x,y} \nonumber \\{} & {} + \left[ \chi + (\psi -\chi ) \delta _{x,L+1}\right] \delta _{y,x-1} , \end{aligned}$$

(3.15)

the decomposition

$$\begin{aligned} \mu ^{x}(\rho ^\star ) = \sum _{y=0}^{L+1} M_{yx} \mu ^{y}(\rho _0), \end{aligned}$$

(3.16)

for $0\le x,y \le L+1$.

Conversely, with an arbitrary constant $\phi$ and with the definition

$$\begin{aligned} \lambda := - \frac{\chi }{1 -\chi } = \frac{\rho _0-\rho ^\star }{\rho _1-\rho ^\star } , \end{aligned}$$

(3.17)

it follows that

$$\begin{aligned} \mu ^{x}(\rho _0) & = \lambda ^x \left[ \mu ^{0}(\rho ^\star ) + (1-\lambda ) \sum _{y=1}^{x} \lambda ^{-y} \mu ^{y}(\rho ^\star ) \right] (1- \delta _{x,L+1}) \nonumber \\ & \quad + \left[ \phi \lambda ^L \left( \mu ^{0}(\rho ^\star ) + (1-\lambda ) \sum _{y=1}^{L} \lambda ^{-y} \mu ^{y}(\rho ^\star ) \right)\right. \nonumber\\ &\quad \left. + (1-\phi ) \mu ^{L+1}(\rho ^\star )\right] \delta _{x,L+1} \nonumber \\ & = \sum _{y=0}^{L+1} \tilde{M} _{yx} \mu ^{y}(\rho ^\star ), \end{aligned}$$

(3.18)

with the matrix elements

$$\begin{aligned} \tilde{M}_{yx}&= {} \left( 1-\lambda + \lambda \delta _{y,0}\right) (1- \frac{\lambda - \phi }{\lambda } \delta _{x,L+1})\\ &\quad \times\left( 1 + \frac{\lambda - \phi }{\phi (1-\lambda )} \delta _{y,L+1}\right) \Theta (x-y) \lambda ^{x-y}, \end{aligned}$$

(3.19)

where $\Theta (z)$ is the Heaviside function. This is a convex combination for $\rho ^\star \le \rho _0$ and $0\le \phi \le 1$. A similar convex combination $\rho ^\star \ge \rho _1$ can be obtained by expanding $\mu ^{x}(\rho _1)$.

To get some intuition for these upper-triangular transformation matrices, we write them explicitly in terms of the parameter $\lambda$, which yields

$$\begin{aligned} M &= \frac{1}{1-\lambda }\\ &\quad \left( \begin{array}{ccccccccc} 1-\lambda &{} -\lambda &{} 0 &{} 0 &{} \dots &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 1 &{} -\lambda &{} 0 &{} \dots &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1 &{} -\lambda &{} \dots &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 1 &{} \dots &{} 0 &{} 0 &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} 1 &{} -\lambda &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} 0 &{} 1 &{} -\lambda &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} 0 &{} 0 &{} 1&{} \psi (1-\lambda ) \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} 0 &{} 0 &{} 0 &{} (1-\psi )(1-\lambda ) \end{array}\right) , \end{aligned}$$

(3.20)

$$\begin{aligned} \tilde{M}&= \left( 1-\lambda \right)\\ &\quad\left( \begin{array}{ccccccccc} \frac{1}{1-\lambda } &{} \frac{\lambda }{1-\lambda } &{} \frac{\lambda ^{2}}{1-\lambda } &{} \frac{\lambda ^{3}}{1-\lambda } &{} \dots &{} \frac{\lambda ^{L-2}}{1-\lambda } &{} \frac{\lambda ^{L-1}}{1-\lambda } &{} \frac{\lambda ^{L}}{1-\lambda } &{} \phi \frac{\lambda ^{L}}{1-\lambda } \\ 0 &{} 1 &{} \lambda &{} \lambda ^2 &{} \dots &{} \lambda ^{L-3} &{} \lambda ^{L-2} &{} \lambda ^{L-1} &{} \phi \lambda ^{L-1} \\ 0 &{} 0 &{} 1 &{} \lambda &{} \dots &{} \lambda ^{L-4} &{} \lambda ^{L-3} &{} \lambda ^{L-2} &{} \phi \lambda ^{L-2} \\ 0 &{} 0 &{} 0 &{} 1 &{} \dots &{} \lambda ^{L-5} &{} \lambda ^{L-4} &{} \lambda ^{L-3} &{} \phi \lambda ^{L-3} \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} 1 &{} \lambda &{} \lambda ^2 &{} \phi \lambda ^{2} \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} 0 &{} 1 &{} \lambda &{} \phi \lambda \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} 0 &{} 0 &{} 1 &{} \phi \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} 0 &{} 0 &{} 0 &{} \frac{1-\phi }{1-\lambda } \end{array}\right) . \end{aligned}$$

(3.21)

Notice that, with $\xi = \phi + \psi (1-\phi ),$

$$\begin{aligned} (M\tilde{M})_{yz} & = \delta _{y,z} - \xi \left( \delta _{y,L+1} - \delta _{y,L}\right) \delta _{z,L+1} \end{aligned}$$

(3.22)

$$\begin{aligned} (\tilde{M}M)_{yz} & = \delta _{y,z} - \xi \lambda ^{L-y} \left( 1-\lambda + \lambda \delta _{y,0} - \delta _{y,L+1} \right) \delta _{z,L+1} . \end{aligned}$$

(3.23)

Hence, $\tilde{M}$ is the inverse of M if and only if $\xi =0$.

3.1.2 Density profile and current

By construction, the shock density profile $\rho ^{x}_k$, i.e., the expectation ${\langle \, {\eta _k}\, \rangle }^{x}$ of the particle number $\eta _k$ in the shock measure $\mu ^{x}$, has a jump discontinuity

$$\begin{aligned} \rho ^{x}_k = \rho _{0} \Theta (x-k-1) + \rho ^\star \delta _{x-k,0} + \rho _1 \Theta (k+1-x) = \left\{ \begin{array}{ll} \displaystyle \rho _0 &{} 1 \le k< x \\ \displaystyle \rho ^\star &{} k=x \\ \displaystyle \rho _1 &{} x < k\le L \end{array} \right. , \end{aligned}$$

(3.24)

at the shock position x for any choice of densities $\rho _1\ne \rho _2$.

The expectation of the particle currents

$$\begin{aligned} j^{x}_k := \left\{ \begin{array}{ll} \displaystyle \alpha {\langle \, {(1-\eta _{1})}\, \rangle }^x - \gamma {\langle \, {\eta _{1}}\, \rangle }^x &{} k = 0 \\ \displaystyle r {\langle \, {\eta _{k}(1-\eta _{k+1})}\, \rangle }^x - \ell {\langle \, {(1-\eta _{k})\eta _{k+1}}\, \rangle }^x &{} k=x \\ \displaystyle \beta {\langle \, {\eta _{L}}\, \rangle }^x - \delta {\langle \, {(1-\eta _{L})}\, \rangle }^x &{} k = L \end{array} \right. , \end{aligned}$$

(3.25)

in a shock measure depends also on the boundary parameters. For densities $\rho _{0,1}$ related by the condition (2.11) for microscopic shock stability and related by

$$\begin{aligned} \rho _0 = \rho _-, \quad \rho _1 = \rho _+, \end{aligned}$$

(3.26)

to the boundary densities $\rho _{\pm }$ appearing in the boundary rates, one finds with

$$\begin{aligned} \begin{array}{ll} j_0 := (r-\ell )\rho _{0}(1-\rho _0), \quad &{} j_1 := (r-\ell )\rho _{1}(1-\rho _1) \\ j^\star _0 := r \rho _{0} (1-\rho ^\star ) - \ell (1-\rho _{0})\rho ^\star , \quad &{} j^\star _1 := r \rho ^\star (1-\rho _{1}) - \ell (1-\rho ^\star ) \rho _{1} \\ j^{\star -}_0 := j^\star _0 + \omega _- (\rho _{0} -\rho ^\star ), \quad &{} j^{\star -}_1 := \omega _- (\rho _{0} - \rho _{1}) \\ j^{\star +}_0 := \omega _+ (\rho _{0} - \rho _{1}), \quad &{} j^{\star +}_1 := j^\star _1 + \omega _+ (\rho ^\star -\rho _{1} ) \end{array}, \end{aligned}$$

(3.27)

the discontinuities

$$\begin{aligned} j^{x}_0 & = j_0 \Theta (x-2) + j^{\star -}_0 \delta _{x-1,0} + j^{\star -}_1 \delta _{x,0}, \end{aligned}$$

(3.28)

$$\begin{aligned} j^{x}_k & = j_0 \Theta (x-k-2) + j^\star _0 \delta _{x-k-1,0} + j^\star _1 \delta _{x-k,0} + j_1 \Theta (k-1-x), \end{aligned}$$

(3.29)

$$\begin{aligned} j^{x}_L & = j^{\star +}_0 \delta _{x-L-1,0} + j^{\star +}_1 \delta _{x-L,0} + j_1 \Theta (L-1-x), \end{aligned}$$

(3.30)

Thus, a single shock measure cannot be stationary even when the nonstationary bulk currents $j_{0,1}$ away from the shock position are equal.

Functions related to the current expectations that appear frequently throughout this work are

$$\begin{aligned} d^{\ell }(\rho _0,\rho _1): & = \frac{j_{0}}{\rho _{1}-\rho _{0}}, \end{aligned}$$

(3.31)

$$\begin{aligned} d^{r}(\rho _0,\rho _1): & = \frac{j_{1}}{\rho _{1}-\rho _{0}}, \end{aligned}$$

(3.32)

$$\begin{aligned} \epsilon ^{\ell }(z): & = \alpha - \gamma z, \end{aligned}$$

(3.33)

$$\begin{aligned} \epsilon ^{r}(z): & = \delta - \beta z . \end{aligned}$$

(3.34)

If the densities $\rho _{0,1}$ are related by microscopic shock stability condition (2.11), then the quantities $d^{\ell ,r}(\rho _0,\rho _1)$ are the microscopic shock hopping rates of a shock in an infinite system [41]. The functions $\epsilon ^{r,\ell }(\rho )$ appear in the boundary currents. For the boundary densities $\rho _\pm$ of the ASEP, one has

$$\begin{aligned} \epsilon ^{\ell }(z_-) = (r-\ell )\rho _-, \quad \epsilon ^{r}(z_+) = - (r-\ell )\rho _+, \end{aligned}$$

(3.35)

where $z_\pm \equiv z(\rho _\pm )$. For $\rho _0=\rho _-$ and $\rho _0=\rho _+$ (which together with (2.11) implies (2.10) with $N=1$), one thus finds the identities

$$\begin{aligned} d^{\ell }(\rho _{0},\rho _{1}) &= \ell \frac{1-\rho _{0}}{1-\rho _{1}} \, = \, r \frac{\rho _{0}}{\rho _{1}} \\ &= \ell (1-\rho _{0}) + r \rho _{0} \, = \, \epsilon ^{\ell }(z_{0}) + \ell , \end{aligned}$$

(3.36)

$$\begin{aligned} d^{r}(\rho _{0},\rho _{1})&= \ell \frac{\rho _{1}}{\rho _{0}} \, = \, r \frac{1-\rho _{1}}{1-\rho _{0}} \\ &= r (1-\rho _{1}) + \ell \rho _1 \, = \, \epsilon ^{r}(z_1) + r , \end{aligned}$$

(3.37)

which also imply $d^{r}(\rho _{0},\rho _{1}) d^{\ell }(\rho _{0},\rho _{1}) = r\ell$ independently of the densities $\rho _{0,1}$. These relations are used frequently throughout this work.

3.2 Matrix product measure

It was shown in [37] that the stationary distribution $\mu ^*(\varvec{\eta })$ of the ASEP defined in the “Introduction” can be written with matrices D, E and boundary vectors ${\langle \, {W}\, |},{| \, {V}\, \rangle }$ in the matrix product form

$$\begin{aligned} \mu ^*(\varvec{\eta }) = \frac{1}{Z_L}{\langle \, {W}\, |}\prod _{k=1}^L (E(1-\eta _k) + D\eta _k){| \, {V}\, \rangle }, \end{aligned}$$

(3.38)

with the partition function

$$\begin{aligned} Z_L {\langle \, {W}\, |}(E + D)^L{| \, {V}\, \rangle }, \end{aligned}$$

(3.39)

which ensures normalization of the stationary distribution. The product is understood to be an ordered product since the matrices D and E do not commute. They satisfy the quadratic algebra

$$\begin{aligned} r DE - \ell ED = s(D+E), \end{aligned}$$

(3.40)

and act on the boundary vectors as

$$\begin{aligned} 0 = {\langle \, {W}\, |}(\alpha E - \gamma D -s), \quad (\beta D - \delta E - s) {| \, {V}\, \rangle } = 0. \end{aligned}$$

(3.41)

Here, s is an arbitrary parameter.

3.2.1 Two-dimensional representations of the MPM

It is straightforward to check that, with arbitrary $s\ne 0$, arbitrary constants $\sigma _{0,1}$ not simultaneously equal to 0, and

$$\begin{aligned} z_+ = q^2 z_-, \end{aligned}$$

(3.42)

to satisfy the constraint (2.10) on the boundary rates with $N=1$, a family of representations is given by

$$\begin{aligned} E(s,\sigma _0) & = \frac{s}{r-\ell } \left( \begin{array}{cc} 1+ z_+^{-1} &{} 0 \\ \sigma _0 &{} 1+ z_-^{-1} \end{array}\right) , \end{aligned}$$

(3.43)

$$\begin{aligned} D(s,\sigma _1) & = \frac{s}{r-\ell } \left( \begin{array}{cc} 1+ z_+ &{} 0 \\ \sigma _1 &{} 1+ z_- \end{array}\right) . \end{aligned}$$

(3.44)

The corresponding boundary vectors are independent of s and can be chosen as

$$\begin{aligned} {\langle \, {W(\sigma _0,\sigma _1)}\, |} = (w(\sigma _0,\sigma _1),1), \quad {| \, {V(\sigma _0,\sigma _1)}\, \rangle } = \left( \begin{array}{c} 1 \\ v(\sigma _0,\sigma _1) \end{array}\right) , \end{aligned}$$

(3.45)

with

$$\begin{aligned} w(\sigma _0,\sigma _1) &= \frac{q^2}{q^2-1} \frac{\sigma _0 z_- + \sigma _1 \kappa _-(\alpha ,\gamma )}{1 - q^2 z_- \kappa _-(\alpha ,\gamma )} \\ &= \frac{(\gamma \sigma _1 - \alpha \sigma _0)\rho _+(1-\rho _+)}{\alpha (1- \rho _+) - \gamma \rho _+ - j_+}, \end{aligned}$$

(3.46)

$$\begin{aligned} v(\sigma _0,\sigma _1) &= \frac{q^{-2}}{1 - q^{-2}} \frac{\sigma _0 z_+ + \sigma _1 \kappa ^{-1}_-(\beta ,\delta )}{1- q^{-2} z_+ \kappa ^{-1}_-(\beta ,\delta )} \\ &= \frac{(\delta \sigma _0 - \beta \sigma _1)\rho _-(1-\rho _-)}{\beta \rho _- - \delta (1- \rho _-) - j_-}. \end{aligned}$$

(3.47)

To make contact with the two-dimensional representations

$$\begin{aligned}{} & {} E_{MS} = E(r-\ell ,1), \quad D_{MS} = D(r-\ell ,0),, \end{aligned}$$

(3.48)

$$\begin{aligned}{} & {} {\langle \, {W}\, |}_{MS} = {\langle \, {W(1,0)}\, |}, \quad {| \, {V}\, \rangle }_{MS} = {| \, {V(1,0)}\, \rangle }, \end{aligned}$$

(3.49)

found in [40] for the bulk hopping rates $r=1$ and $\ell =x$, we introduce the transformation matrix

$$\begin{aligned} \Gamma = \left( \begin{array}{cc} \sigma _0 + \frac{\sigma _1}{z_1z_0} &{} 0 \\ \frac{\sigma _1}{z_0-z_1} &{} 1 \end{array}\right) . \end{aligned}$$

(3.50)

With the choice $s = r-\ell$, a short computation yields

$$\begin{aligned} E_{MS} & = \Gamma E(r-\ell ,\sigma _0) \Gamma ^{-1} \end{aligned}$$

(3.51)

$$\begin{aligned} D_{MS} & = \Gamma D(r-\ell ,\sigma _1) \Gamma ^{-1}, \end{aligned}$$

(3.52)

$$\begin{aligned} {\langle \, {W}\, |}_{MS} & = {\langle \, {W(\sigma _0,\sigma _1)}\, |} \Gamma ^{-1} , \end{aligned}$$

(3.53)

$$\begin{aligned} {| \, {V}\, \rangle }_{MS} & = \Gamma {| \, {V(\sigma _0,\sigma _1)}\, \rangle } . \end{aligned}$$

(3.54)

Thus, the representation of [40] and the generic representation (3.43) and (3.44) are related by the similarity transformation (3.50).¹

3.2.2 Relation between the MPM and shock measures

From now on, we take

$$\begin{aligned} s=(r-\ell )\frac{z_-}{(1+ z_-)^2} = (r-\ell )\rho _-(1-\rho _-), \end{aligned}$$

(3.55)

and keep $\sigma _0$ and $\sigma _1$ general. Skipping the dependence of the representation matrices and boundary vectors on the parameters $s,\sigma _{0,1}$, one gets

$$\begin{aligned} E & = (1- \rho _-) \left( \begin{array}{cc} \frac{\rho _-}{\rho _+} &{} 0 \\ \sigma _0 \rho _- &{} 1 \end{array}\right) , \end{aligned}$$

(3.56)

$$\begin{aligned} D & = \rho _- \left( \begin{array}{cc} \frac{1-\rho _-}{1-\rho _+} &{} 0 \\ \sigma _1 (1-\rho _-) &{} 1 \end{array}\right) , \end{aligned}$$

(3.57)

in terms of the boundary densities $\rho _\pm$.

To obtain the relationship to shock measures, we choose $\rho _0=\rho _-$, $\rho _1=\rho _+$, which with the microscopic shock stability condition (2.11) guarantees that the constraint (2.10) on the boundary rates is satisfied for $N=1$. Furthermore, we define the spin-1/2 ladder operators

$$\begin{aligned} \sigma ^+ := \left( \begin{array}{cc} 0 &\quad 1 \\ 0 &\quad 0 \end{array}\right) , \quad \sigma ^- := \left( \begin{array}{cc} 0 &{} 0 \\ 1 &{} 0 \end{array}\right) , \end{aligned}$$

(3.58)

and the projectors

$$\begin{aligned} \hat{v} := \left( \begin{array}{cc} 1 &\quad 0 \\ 0 &\quad 0 \end{array}\right) , \quad \hat{n} := \left( \begin{array}{cc} 0 &{} 0 \\ 0 &{} 1 \end{array}\right) , \end{aligned}$$

(3.59)

and we introduce the linear combination

$$\begin{aligned} S_k(\sigma _0,\sigma _1) = \frac{\sigma _0 (1-\eta _k) + \sigma _1 \eta _k}{r-\ell } \end{aligned}$$

(3.60)

of Bernoulli distributions $B_k$ (3.5). With

$$\begin{aligned} X_k := E(1-\eta _k) + D\eta _k = \frac{j_0}{j_1} B_k(\rho _1) \hat{v} + B_k(\rho _0) \hat{n} + j_0 S_k(\sigma _0,\sigma _1) \sigma ^-, \end{aligned}$$

(3.61)

and the choice $\sigma _0 = (r-\ell ) (1-\rho ^\star )$ and $\sigma _1 = (r-\ell ) \rho ^\star$ (so that $S_k(\sigma _0,\sigma _1) = B_k(\rho ^\star )$), the matrix product appearing in the MPM (3.38) becomes

$$\begin{aligned} X_1 \dots X_L = \left( \frac{j_0}{j_1}\right) ^L \mu ^{0}(\rho ^\star ) \hat{v} + j_0 \sum _{x=1}^L \left( \frac{j_0}{j_1}\right) ^{L-x} \mu ^{x}(\rho ^\star ) \sigma ^- + \mu ^{L+1}(\rho ^\star ) \hat{n} . \end{aligned}$$

(3.62)

This follows from the properties $\hat{v} \sigma ^- = 0$, $\sigma ^- \hat{v} = \sigma ^-$ $\hat{n} \sigma ^- = \sigma ^-$, and $\sigma ^- \hat{n} = 0$ of the spin-1/2 operators.

Since

$$\begin{aligned} {\langle \, {W}\, |} \hat{v} {| \, {V}\, \rangle } = w, \quad {\langle \, {W}\, |} \hat{n} {| \, {V}\, \rangle } = v, \quad {\langle \, {W}\, |} \sigma ^- {| \, {V}\, \rangle } = 1, \end{aligned}$$

(3.63)

with

$$\begin{aligned} w & = \frac{\alpha (1-\rho ^\star ) - \gamma \rho ^\star }{ j_1 - \alpha (1- \rho _1) + \gamma \rho _1} j_1 \nonumber \\ & = \frac{j^{1}_0}{j^{0}_1 - j^{0}_0} j_1, \end{aligned}$$

(3.64)

$$\begin{aligned} v & = \frac{\beta \rho ^\star - \delta (1-\rho ^\star )}{ j_0 - \beta \rho _0 + \delta (1- \rho _0)} j_0 \, = \, \frac{j^{L}_L}{j^{L+1}_{L-1}- j^{L+1}_{L}} j_0, \end{aligned}$$

(3.65)

we arrive at

$$\begin{aligned} \mu ^*(\varvec{\eta }) = \frac{1}{Z_L} \left[ w \left( \frac{j_0}{j_1}\right) ^L \mu ^{0}(\rho ^\star ) + j_0 \sum _{x=1}^L \left( \frac{j_0}{j_1}\right) ^{L-x} \mu ^{x}(\rho ^\star ) + v \mu ^{L+1}(\rho ^\star ) \right] , \end{aligned}$$

(3.66)

with the normalization

$$\begin{aligned} Z_L = w \left( \frac{j_0}{j_1}\right) ^L + v + j_0 \frac{1-\left( \frac{j_0}{j_1}\right) ^L}{1-\frac{j_0}{j_1}}. \end{aligned}$$

(3.67)

We conclude that varying the shock densities $\rho ^\star$ in the shock measures corresponds to a similarity transformation of the two-dimensional representation. Since expectation values are independent of similarity transformations, they are also independent of the choice of shock density. The representation matrices can be expressed entirely in terms of the densities $\rho _\pm$ and $\rho ^\star$ characterizing the BSM, and the boundary vectors can be expressed entirely in terms of the local shock currents (3.27).

4 Reverse duality

This section contains the main result of this work, which is an intertwining duality for the ASEP on the parameter manifold defined by 2.10 with $N=1$ and the resulting explicit expression for the time evolution of shock measure with a single shock. First, we define the open ASEP and introduce our notation (Sect. 4.1) and provide a mathematical toolbox for the derivation of the duality (Sect. 4.2). This derivation is done in Sect. 4.3. It is first shown that the probability vectors corresponding to the shock measures form an invariant subspace under the action of the generator H. The coefficients appearing in this action are computed using the toolbox and turn out to be the transition rates of a random walk. This implies duality and consequently the shock random walk described in the “Introduction.” Depending on the choice of configuration at the shock position, one obtains either reflecting boundaries or absorbing boundaries. The relationship between these two cases is then shown to be a Siegmund duality for random walks. The section concludes with some remarks on the spectrum of the generator.

4.1 Generator of the ASEP with open boundaries

To derive the intertwining duality that yields the shock random walk, we follow standard procedure [77] and represent the generator of the ASEP in terms of the quantum Hamiltonian H of the XXZ Heisenberg quantum spin chain with nondiagonal boundary fields [78]. In this approach, a probability distribution $P_t(\varvec{\eta })$ is represented by a probability vector

$$\begin{aligned} {| \, {P(t)}\, \rangle } = \sum _{\varvec{\eta }} P_t(\varvec{\eta }) {| \, {\varvec{\eta }}\, \rangle } ,\end{aligned}$$

(4.1)

where the ket-vectors ${| \, {\varvec{\eta }}\, \rangle }$ form a canonical tensor basis of the vector space $({\mathbb C}^2)^{\otimes L}$ endowed with the standard scalar product ${\langle \, {\varvec{\eta }} \, | \, {\varvec{\eta }'} \, \rangle }=\delta _{\varvec{\eta },\varvec{\eta }'}$. For a single site we choose the basis

$$\begin{aligned} {| \, {0}\, \rangle } = \left( \begin{array}{c} 1 \\ 0 \end{array}\right) , \quad {| \, {1}\, \rangle } = \left( \begin{array}{c} 0 \\ 1 \end{array}\right) , \end{aligned}$$

(4.2)

so that the single-site Bernoulli measure $B(\rho )$ with density $\rho$ is represented by the two-dimensional vector

$$\begin{aligned} {| \, {\rho }\, \rangle } = (1-\rho ) {| \, {0}\, \rangle } + \rho {| \, {1}\, \rangle }. \end{aligned}$$

(4.3)

Analogously, ${| \, {abc\dots }\, \rangle }$ denotes the Kronecker product ${| \, {a}\, \rangle }\otimes {| \, {b}\, \rangle }\otimes {| \, {c}\, \rangle }\otimes \dots$ of two-dimensional vectors ${| \, {a}\, \rangle },{| \, {b}\, \rangle },{| \, {c}\, \rangle }\dots$ with components $(a_0,a_1), (b_0,b_1), (c_0,c_1), \dots$. A function $f(\varvec{\eta })$ of the configurations $\varvec{\eta }$ is represented by the diagonal matrix

$$\begin{aligned} \hat{f} = \sum _{\varvec{\eta }} f(\varvec{\eta }) {| \, {\varvec{\eta }}\, \rangle }{\langle \, {\varvec{\eta }}\, |}. \end{aligned}$$

(4.4)

The Markovian time evolution is expressed by the master equation

$$\begin{aligned} \frac{\textrm{d}}{\textrm{d}t}{| \, {P(t)}\, \rangle } = - H {| \, {P(t)}\, \rangle }. \end{aligned}$$

(4.5)

See [7] for further details.

With the matrices

$$\begin{aligned} \tilde{h}_{k,k+1} & = r (\hat{n}_{k} \hat{v}_{k+1} - \sigma ^+_{k} \sigma ^-_{k+1}) + \ell (\hat{v}_{k} \hat{n}_{k+1} - \sigma ^-_{k} \sigma ^+_{k+1})\\ &\quad + (r-\ell ) \left( \hat{n}_{k+1} - \hat{n}_{k}\right) \end{aligned}$$

(4.6)

$$\begin{aligned} \tilde{h}^{-}_{1} & = \alpha (\hat{v}_{1} - \sigma ^-_{1}) + \gamma (\hat{n}_{1} - \sigma ^+_{1}) + (r-\ell ) \hat{n}_{1} \end{aligned}$$

(4.7)

$$\begin{aligned} \tilde{h}^{+}_{L} & = \delta (\hat{v}_{L} - \sigma ^-_{L}) + \beta (\hat{n}_{L} - \sigma ^+_{L}) + (r-\ell ) \hat{n}_{L}, \end{aligned}$$

(4.8)

and

$$\begin{aligned} \tilde{H}_{bulk} = \sum _{k=1}^{L-1} \tilde{h}_{k,k+1}, \end{aligned}$$

(4.9)

the quantum Hamiltonian form of the generator of the open ASEP is given by

$$\begin{aligned} H = \tilde{H}_{bulk} + \tilde{h}^{-}_{1} + \tilde{h}^{+}_{L}. \end{aligned}$$

(4.10)

The BSM $\mu ^x_{\varvec{\eta }}(\rho ^\star )$ with shock at position x is given by the Kronecker product

$$\begin{aligned} {| \, {\mu ^{x}(\rho ^\star )}\, \rangle } = \sum _{\varvec{\eta }} \mu ^x_{\varvec{\eta }}(\rho ^\star ) {| \, {\varvec{\eta }}\, \rangle } = {| \, {\rho _{0}}\, \rangle }^{x-1} \otimes {| \, {\rho ^\star }\, \rangle } \otimes {| \, {\rho _{1}}\, \rangle }^{\otimes (L-x-1)}, \end{aligned}$$

(4.11)

with the conventions $A^{\otimes 0} = 1$, $A^{\otimes 1} = A$ for any matrix or vector A.

4.2 Toolbox

In a slight abuse of language, we call for any two-dimensional vector ${| \, {a}\, \rangle } = a_0 {| \, {0}\, \rangle } + a_1 {| \, {1}\, \rangle }$ with $a_0\ne 0$ the component ratio

$$\begin{aligned} z_a := \frac{a_1}{a_0}, \end{aligned}$$

(4.12)

the fugacity and for two-dimensional vectors ${| \, {a}\, \rangle }, {| \, {b}\, \rangle }$ we define the determinant

$$\begin{aligned} \Delta _{ab} := a_0 b_1 - a_1 b_0, \end{aligned}$$

(4.13)

which vanishes if and only if the two vectors are linearly dependent and that for $z_b = q^2 z_a$ has the property

$$\begin{aligned} \Delta _{ab} = (r-\ell ) \frac{a_0 b_1}{r} = (r-\ell ) \frac{a_1 b_0}{\ell }. \end{aligned}$$

(4.14)

For linearly independent vectors, we also define the functions

$$\begin{aligned}{} & {} d(a,b) := (r-\ell )\frac{a_1a_0}{\Delta _{ab}}, \quad \tilde{d}(a,b) := (r-\ell )\frac{a_1b_0}{\Delta _{ab}}, \end{aligned}$$

(4.15)

$$\begin{aligned}{} & {} d_\pm (a,b) := \frac{\epsilon ^{r,\ell }(z_a) a_0 (a_0+a_1)}{\Delta _{ab}}, \\ &\quad \tilde{d}_\pm (a,b) := \frac{\epsilon ^{r,\ell }(z_a) a_0 (b_0+b_1)}{\Delta _{ab}} . \end{aligned}$$

(4.16)

In [25], the following was proved:

• (A) For vectors ${| \, {\rho _\pm }\, \rangle }$ with fugacities such that $z_- \equiv z(\rho _-) = \kappa ^{-1}_+(\alpha ,\gamma )$ and $z_+\equiv z(\rho _+) = \kappa _+(\beta ,\delta )$:

  $$\begin{aligned} \tilde{h}^{-} {| \, {\rho _-}\, \rangle } = \epsilon ^{\ell }(z_-) {| \, {\rho _-}\, \rangle } , \quad \tilde{h}^{+} {| \, {\rho _+}\, \rangle } = \epsilon ^{r}(z_+) {| \, {\rho _+}\, \rangle } . \end{aligned}$$

  (4.17)

• (B) For any pair of linearly independent vectors ${| \, {a}\, \rangle }$ and ${| \, {b}\, \rangle }$:

  $$\begin{aligned} \tilde{h}^{\pm } {| \, {a}\, \rangle } & = [\tilde{d}_\pm (a,b) \pm \tilde{d}(a,b)] {| \, {a}\, \rangle } - [d_\pm (a,b) \pm d(a,b)] {| \, {b}\, \rangle }. \end{aligned}$$

  (4.18)

• (C) For any pair of vectors:

  $$\begin{aligned} \tilde{h} {| \, {ab}\, \rangle } & = \Delta _{ab} \left( r {| \, {01}\, \rangle } - \ell {| \, {10}\, \rangle }\right) . \end{aligned}$$

  (4.19)

• (D) For vectors ${| \, {a^{(i)}}\, \rangle }$, ${| \, {\tilde{a}^{(i)}}\, \rangle }$, $i\in \{1,2\}$, such that $a^{(i)}_0 a^{(i)}_1 \ne 0$, $\tilde{a}^{(i)}_0 \tilde{a}^{(i)}_1 \ne 0$, and

  $$\begin{aligned} \frac{z(\tilde{a}^{(i)})}{z(a^{(i)})} = q^2 , \end{aligned}$$

  (4.20)

  and a pair of linearly independent vectors (${| \, {b^{(i)}}\, \rangle }$, ${| \, {\tilde{b}^{(i)}}\, \rangle }$) such that ${| \, {b^{(1)}}\, \rangle }$ is linearly independent of ${| \, {\tilde{a}^{(1)}}\, \rangle }$ and ${| \, {b^{(2)}}\, \rangle }$ is linearly independent of ${| \, {\tilde{a}^{(2)}}\, \rangle }$:

  $$\begin{aligned} &\tilde{h} {| \, {a^{(1)}b^{(1)}}\, \rangle }\\ &\quad = u^{(1)} {| \, {a^{(1)}b^{(1)}}\, \rangle } - v^{(1)} {| \, {b^{(1)}\tilde{a}^{(1)}}\, \rangle } - w^{(1)} {| \, {a^{(1)}\tilde{b}^{(1)}}\, \rangle }, \end{aligned}$$

  (4.21)
  $$\begin{aligned} &\tilde{h} {| \, {b^{(2)}\tilde{a}^{(2)}}\, \rangle }\\ &\quad = u^{(2)} {| \, {b^{(2)}\tilde{a}^{(2)}}\, \rangle } - v^{(2)} {| \, {a^{(2)}b^{(2)}}\, \rangle } - w^{(2)} {| \, {\tilde{b}^{(2)}\tilde{a}^{(2)}}\, \rangle }, \end{aligned}$$

  (4.22)

  with coefficients

  $$\begin{aligned}{} & u^{(1)} = \ell - \tilde{d}(b^{(1)},\tilde{b}^{(1)}), \quad v^{(1)} = d(a^{(1)},\tilde{a}^{(1)}),\\ & w^{(1)} = - d(b^{(1)},\tilde{b}^{(1)}), \end{aligned}$$

  (4.23)
  $$\begin{aligned} & u^{(2)} = r + \tilde{d}(b^{(2)}, \tilde{b}^{(2)}), \quad v^{(2)} = - d(\tilde{a}^{(2)}, a^{(2)}),\\ & w^{(2)} = d(b^{(2)}, \tilde{b}^{(2)}). \end{aligned}$$

  (4.24)

4.3 Time evolution of a shock measure

As pointed out in the “Introduction,” intertwining duality can be expressed as a reverse duality (2.4) for reversible dual processes. We prove an intertwining duality with a duality matrix S with elements

$$\begin{aligned} S_{\varvec{\eta }x}(\sigma _0,\sigma _1) & = {\langle \, {\varvec{\eta }} \, | \, {\mu ^x(\sigma _0,\sigma _1)} \, \rangle } = \mu ^x_{\varvec{\eta }}(\sigma _0,\sigma _1), \end{aligned}$$

(4.25)

$$\begin{aligned} & = A_L(\sigma _0,\sigma _1) \left( q\sqrt{\frac{j_0}{j_1}}\right) ^{x-1} z_1^{N(\varvec{\eta })} q^{-2N_{x-1}(\varvec{\eta })} S_x(\sigma _0,\sigma _1) , \end{aligned}$$

(4.26)

given by the BSMs defined in analogy to (3.6) with $z_1 = q^2 z_0$ but with $B_x(\rho ^\star )$ replaced by the general linear combination $S_x(\sigma _0,\sigma _1)$ defined in (3.60). The normalization $A_L(\sigma _0,\sigma _1)$ is irrelevant from the viewpoint of duality but will be taken to be equal to the normalization $A_L(\rho ^\star )$ in (3.9) for the choice $\sigma _0 = (r-\ell ) (1-\rho ^\star )$, $\sigma _1 = (r-\ell ) \rho ^\star$ that yields the normalized BSM $S_{\varvec{\eta }x}(\sigma _0,\sigma _1) = \mu ^x_{\varvec{\eta }}(\rho ^\star )$. For arbitrary values of $\sigma _{0,1}$, the functions $\mu ^x_{\varvec{\eta }}(\sigma _{0},\sigma _{1})$ defined analogously to (3.6) but with a generic function $S_k(\sigma _{0},\sigma _{1})$, one obtains the linear combination

$$\begin{aligned} \mu ^x_{\varvec{\eta }}(\sigma _{0},\sigma _{1}) := \frac{\sigma _0 \mu ^x_{\varvec{\eta }}(0) + \sigma _1 \mu ^x_{\varvec{\eta }}(1)}{r-\ell }, \end{aligned}$$

(4.27)

of BSMs with extremal shock densities $\rho ^\star =0$ and $\rho ^\star =1$ respectively. Below, we omit the dependence on $\sigma _0,\sigma _1$ or $\rho ^\star$ respectively, when no confusion can arise.

Notice that S is the matrix with the column vectors ${| \, {\mu ^{x}}\, \rangle }$ as rows. The intertwining duality (2.4) thus implies that the vectors ${| \, {\mu ^{x}}\, \rangle }$ form an invariant subspace under the action of the quantum Hamiltonian, i.e.,

$$\begin{aligned} H {| \, {\mu ^{x}}\, \rangle } = \sum _{y=0}^{L+1} G_{yx} {| \, {\mu ^{y}}\, \rangle }. \end{aligned}$$

(4.28)

and it turns out that G is the generator of a biased random walk. This duality then implies the random walk property (2.6) of the shock position. In [17], this was proved for a particular shock density and the special case of boundary parameters satisfying in addition to (2.10) for general N the extra constraint (2.13). Here we take $N=1$ only but drop the special condition. Moreover, we allow for vectors ${| \, {\mu ^{\textbf{y}}}\, \rangle }$ given by the function $S_k(\sigma _0,\sigma _1)$ without constraint on the parameters $\sigma _{0,1}$ except that trivially $\sigma _0=\sigma _1=0$ has to be excluded.

We point out the following:

• For $\sigma _0 + \sigma _1 \ne 0$, one can choose without loss of generality $\sigma _0 + \sigma _1 = r-\ell$ since the value of $\sigma _0+\sigma _1$ can be absorbed in the normalization of the measure. For $\sigma _0 = (r-\ell )(1-\rho ^\star )$ and $\sigma _1 = (r-\ell )\rho ^\star$, this yields functions $\mu ^x_{\varvec{\eta }}(\rho ^\star )$ of the form (3.6) but without the convexity condition $0 \le \rho ^\star \le 1$ that makes $\mu ^x_{\varvec{\eta }}(\rho ^\star )$ a probability measure.

• We recall that, for $\rho ^\star = \rho _0$, one has ${| \, {\mu ^{L+1}(\rho _{0})}\, \rangle } = {| \, {\mu ^{L}(\rho _{0})}\, \rangle }$ while for $\rho ^\star = \rho _1$ one has ${| \, {\mu ^{0}(\rho _{1})}\, \rangle } = {| \, {\mu ^{1}(\rho _{1})}\, \rangle }$. This linear dependence of boundary shock measures requires treating the cases $\rho ^\star \in \{\rho _0,\rho _1\}$ separately.

• In the last remaining case $\sigma _0 + \sigma _1 = 0$, one can choose without loss of generality $\sigma _0 = r-\ell$ since the value of $\sigma _0$ can be absorbed in the normalization of the difference measure

  $$\begin{aligned} \theta ^x_{\varvec{\eta }} := \mu ^x_{\varvec{\eta }}(0) - \mu ^x_{\varvec{\eta }}(1), \end{aligned}$$

  (4.29)

  resulting from (4.27).

In short, one needs to consider only the functions $\mu ^x_{\varvec{\eta }}(\rho ^\star )$ (without constraint on the range of $\rho ^\star$) and $\theta ^x(\varvec{\eta })$ and it remains (i) to prove that the invariance property (4.28) is valid for an arbitrary choice of $\sigma _{0,1}$, (ii) to compute the matrix elements of G, and (iii) to show under which conditions they define the generator of a random walk dynamics with positive rates.

4.3.1 Dual random walk with reflecting boundaries

We take $\sigma _0 = (r-\ell )(1-\rho ^\star )$ and $\sigma _1 = (r-\ell )\rho ^\star$ and consider the functions $\mu ^x_{\varvec{\eta }}(\rho ^\star )$ for arbitrary value of the parameter $\rho ^\star$. Nevertheless, we refer to x as shock position even when $\rho ^\star \notin [0,1]$. The first step is to prove the invariance of the subspace spanned by the vectors ${| \, {\mu ^{x}(\rho ^\star )}\, \rangle }$. To make the notation less heavy, the dependence of the shock measure on $\rho ^\star$ will be dropped.

Shock in the range $1< x < L$: From the product form of the shock measure, the eigenvalue properties (4.19) and (4.17) and the conditions (2.10) for $N=1$ and (2.11), one concludes

$$\begin{aligned} H {| \, {\mu ^{x}}\, \rangle } & = (\tilde{h}^{-}_{1} + \tilde{h}_{x-1,x} +\tilde{h}_{x,x+1} + \tilde{h}^{+}_{L}) {| \, {\mu ^{x}}\, \rangle } \nonumber \\ & = {| \, {\rho _{0}}\, \rangle }^{\otimes (x - 1 -1)}\otimes [(h_{12} + h_{23}) {| \, {\rho _{0}}\, \rangle } \otimes {| \, {\rho ^\star }\, \rangle }\\ &\quad \otimes {| \, {\rho _{1}}\, \rangle }] \otimes {| \, {\rho _{1}}\, \rangle }^{\otimes (L - x -1)} \nonumber \\{} & {} + (\epsilon ^{\ell }(\rho _0) + \epsilon ^{r}(\rho _1)) {| \, {\mu ^{x}}\, \rangle }, \end{aligned}$$

(4.30)

with the functions (3.33) and (3.34). Relations (4.21) and (4.22) yield after taking Kronecker products on the right in (4.21) and on the left in (4.22)

$$\begin{aligned} h_{12}{| \, {\rho _{0}\rho ^\star \rho _1}\, \rangle } & = \left( \ell - \tilde{d}(\rho ^\star ,\tau _1)\right) {| \, {\rho _{0}\rho ^\star \rho _1}\, \rangle } \nonumber \\{} & {} - d(\rho _{0},\rho _{1}) {| \, {\rho ^\star \rho _{1}\rho _1}\, \rangle } + d(\rho ^\star ,\tau _{1}) {| \, {\rho _{0}\tau _{1}\rho _1}\, \rangle }, \end{aligned}$$

(4.31)

$$\begin{aligned} h_{23} {| \, {\rho _{0}\rho ^\star \rho _{1}}\, \rangle } & = \left( r + \tilde{d}(\rho ^\star ,\tau _{3}) \right) {| \, {\rho _{0}\rho ^\star \rho _{1}}\, \rangle } \nonumber \\{} & {} + d(\rho _{1},\rho _{0}) {| \, {\rho _{0}\rho _{0}\rho ^\star }\, \rangle } - d(\rho ^\star ,\tau _{3}) {| \, {\rho _{0}\tau _{3}\rho _{1}}\, \rangle }, \end{aligned}$$

(4.32)

with the functions (4.15) and arbitrary vectors ${| \, {\tau _1}\, \rangle }$ and ${| \, {\tau _3}\, \rangle }$ (except for the immaterial linear independence ${| \, {\rho ^\star }\, \rangle }$). Choosing ${| \, {\tau _1}\, \rangle } = {| \, {\tau _3}\, \rangle }$ leads to a cancelation of all terms involving ${| \, {\tau _1}\, \rangle }$ and ${| \, {\tau _3}\, \rangle }$, and the local invariance property

$$\begin{aligned} (h_{12} + h_{23}){| \, {\rho _0\rho ^\star \rho _1}\, \rangle } = (r+\ell ) {| \, {\rho _0\rho ^\star \rho _1}\, \rangle } - d(\rho _{0},\rho _{1}) {| \, {\rho ^\star \rho _1\rho _1}\, \rangle } + d(\rho _{1},\rho _{0}) {| \, {\rho _0\rho _0\rho ^\star }\, \rangle }, \end{aligned}$$

(4.33)

follows. Taking Kronecker products with densities $\rho _0$ on the left and $\rho _1$ on the right and using (4.14) together with the identities (3.36) and (3.37) then yields

$$\begin{aligned} H {| \, {\mu ^{x}}\, \rangle } & = [r + \ell + \epsilon ^{\ell }(\rho _0) + \epsilon ^{r}(\rho _1)] {| \, {\mu ^{x}}\, \rangle } - d^{\ell } {| \, {\mu ^{x-1}}\, \rangle } - d^{r} {| \, {\mu ^{x+1}}\, \rangle } \nonumber \\ & = (d^{r} + d^{\ell }) {| \, {\mu ^{x}}\, \rangle } - d^{\ell } {| \, {\mu ^{x-1}}\, \rangle } - d^{r} {| \, {\mu ^{x+1}}\, \rangle } , \end{aligned}$$

(4.34)

with the dependence of the shock hopping rates on the densities $\rho _{0,1}$ dropped. For $1< x < L$, we arrive at

$$\begin{aligned} G_{yx}(\rho ^\star ) & = (d^{\ell } + d^{r}) \delta _{x,y} - d^{\ell } \delta _{x,y+1} - d^{r} \delta _{x,y-1}. \end{aligned}$$

(4.35)

for any $\rho ^\star$.

Shock at the left boundary: ${({\hbox {a}}) \rho ^\star \ne \rho _1:}$ From the eigenvalue properties (4.17) one gets

$$\begin{aligned} H {| \, {\mu ^{1}}\, \rangle } & = (\tilde{h}^{-}_{1} + h_{1,2} + \tilde{h}^{+}_{L}) {| \, {\mu ^{1}}\, \rangle } \nonumber \\ & = [(\tilde{h}^{-}_{1} + h) {| \, {\rho ^\star }\, \rangle }\otimes {| \, {\rho _1}\, \rangle }] \otimes {| \, {\rho _1}\, \rangle }^{\otimes {L-2}} + \epsilon ^{r}(\rho _1) {| \, {\mu ^{1}}\, \rangle }, \end{aligned}$$

(4.36)

$$\begin{aligned} H {| \, {\mu ^{0}}\, \rangle } & = (\tilde{h}^{-} + \tilde{h}^{+}_{L}) {| \, {\mu ^{0}}\, \rangle } \nonumber \\ & = [\tilde{h}^{-} {| \, {\rho _1}\, \rangle }] \otimes {| \, {\rho _1}\, \rangle }^{\otimes {L-1}} + \epsilon ^{r}(\rho _1) {| \, {\mu ^{1}}\, \rangle }. \end{aligned}$$

(4.37)

To handle the left boundary term involving $\tilde{h}$ in (4.36), we use (4.22) and take ${| \, {b^{(2)}}\, \rangle } = {| \, {\rho ^\star }\, \rangle }$ and ${| \, {\tilde{a}^{(2)}}\, \rangle } = {| \, {\rho _1}\, \rangle }$. This yields

$$\begin{aligned} \tilde{h} {| \, {\rho ^\star \rho _{1}}\, \rangle } = r {| \, {\rho ^\star \rho _{1}}\, \rangle } + d(\rho _{1},\rho _{0}) {| \, {\rho _{0}\rho ^\star }\, \rangle } + \tilde{d}(\rho ^\star ,\tau _{3}) {| \, {\rho ^\star \rho _{1}}\, \rangle } - d(\rho ^\star ,\tau _{3}) {| \, {\tau _{3}\rho _{1}}\, \rangle }. \end{aligned}$$

(4.38)

For the boundary term involving $\tilde{h}^{-}$ we use (4.18) and observe that for $a_0 = 1-\rho ^\star$ and $a_1 = \rho ^\star$ one has $\tilde{d}_-(\rho ^\star ,b) = d_-(\rho ^\star ,b)$. Therefore,

$$\begin{aligned} \tilde{h}^{-} {| \, {\rho ^\star \rho _1}\, \rangle } = \left( d_-(\rho ^\star ,b) - \tilde{d}(\rho ^\star ,b)\right) {| \, {\rho ^\star \rho _1}\, \rangle } - \left( d_-(\rho ^\star ,b) - d(\rho ^\star ,b)\right) {| \, {b\rho _1}\, \rangle } , \end{aligned}$$

(4.39)

with vectors ${| \, {\tau _3}\, \rangle }$ and ${| \, {b}\, \rangle }$ that are arbitrary except for the required but immaterial linear independence of ${| \, {\rho ^\star }\, \rangle }$. Choosing ${| \, {\tau _3}\, \rangle } = {| \, {b}\, \rangle } = {| \, {\rho _1}\, \rangle }$ leads to the projection property

$$\begin{aligned} (\tilde{h}^{-}_1 + h) {| \, {\rho ^\star \rho _{1}}\, \rangle } = \left( r + d_-(\rho ^\star ,\rho _1) \right) {| \, {\rho ^\star \rho _{1}}\, \rangle } + d(\rho _{1},\rho _{0}) {| \, {\rho _{0}\rho ^\star }\, \rangle } - d_-(\rho ^\star ,\rho _1) {| \, {\rho _1\rho _{1}}\, \rangle }, \end{aligned}$$

(4.40)

and using (3.37) one obtains

$$\begin{aligned} H {| \, {\mu ^{1}}\, \rangle } & = \left( d^{r} + d_-(\rho ^\star ,\rho _1) \right) {| \, {\mu ^{1}}\, \rangle } - d^{r} {| \, {\mu ^{2}}\, \rangle } - d_-(\rho ^\star ,\rho _1) {| \, {\mu ^{0}}\, \rangle } . \end{aligned}$$

(4.41)

To treat (4.37) we take in (4.18) ${| \, {{| \, {\tau _1}\, \rangle }}\, \rangle }={| \, {\rho _1}\, \rangle }$ and ${| \, {b}\, \rangle } = {| \, {\rho ^\star }\, \rangle }$ which yields

$$\begin{aligned} \tilde{h}^{-} {| \, {\rho _1}\, \rangle } = \left( d_-(\rho _1,\rho ^\star ) - \tilde{d}(\rho _1,\rho ^\star )\right) {| \, {\rho _1}\, \rangle } - \left( d_-(\rho _1,\rho ^\star ) - d(\rho _1,\rho ^\star )\right) {| \, {\rho ^\star }\, \rangle }, \end{aligned}$$

(4.42)

and therefore

$$\begin{aligned} H {| \, {\mu ^{0}}\, \rangle } & = \left( d_-(\rho _1,\rho ^\star ) - d(\rho _1,\rho ^\star )\right) \left( {| \, {\mu ^{0}}\, \rangle } - {| \, {\mu ^{1}}\, \rangle } \right) . \end{aligned}$$

(4.43)

Hence with

$$\begin{aligned} d^{\ell }_{-}(\rho ^\star ): & = d_-(\rho ^\star ,\rho _1), \end{aligned}$$

(4.44)

$$\begin{aligned} d^{r}_{-}(\rho ^\star ): & = d_-(\rho _1,\rho ^\star ) - d(\rho _1,\rho ^\star ) , \end{aligned}$$

(4.45)

one gets

$$\begin{aligned} G_{y1}(\rho ^\star ) & = (d^{\ell }_{-}(\rho ^\star ) + d^{r}) \delta _{y,1} - d^{\ell }_{-}(\rho ^\star ) \delta _{y,0} - d^{r} \delta _{y,2}, \end{aligned}$$

(4.46)

$$\begin{aligned} G_{y0}(\rho ^\star ) & = d^{r}_{-}(\rho ^\star ) \left( \delta _{y,0} - \delta _{y,1}\right) , \end{aligned}$$

(4.47)

for $\rho ^\star \ne \rho _1$.

${({\hbox {b}})\rho ^\star =\rho _1:}$ The strategy is analogous to the general case treated above. We choose in (4.18) ${| \, {a}\, \rangle }={| \, {\rho _1}\, \rangle }$ and ${| \, {b}\, \rangle } = {| \, {\rho _0}\, \rangle }$, which gives

$$\begin{aligned} \tilde{h}^{-} {| \, {\rho _1}\, \rangle } = \left( d_-(\rho _1,\rho _0) - \tilde{d}(\rho _1,\rho _0)\right) {| \, {\rho _1}\, \rangle } - \left( d_-(\rho _1,\rho _0) - d(\rho _1,\rho _0)\right) {| \, {\rho _0}\, \rangle } . \end{aligned}$$

(4.48)

Thus with (3.37)

$$\begin{aligned} H {| \, {\mu ^{1}}\, \rangle } & = \left( d_-(\rho _1,\rho _0) - d(\rho _1,\rho _0)\right) \left( {| \, {\mu ^{1}}\, \rangle } - {| \, {\mu ^{2}}\, \rangle }\right) . \end{aligned}$$

(4.49)

and with

$$\begin{aligned} \tilde{d}^{r}_{-} := d_-(\rho _1,\rho _0) - d(\rho _1,\rho _0), \end{aligned}$$

(4.50)

one gets

$$\begin{aligned} G_{y1}(\rho ^\star ) = \tilde{d}^{r}_{-} \left( \delta _{y,1} - \delta _{y,2}\right) . \end{aligned}$$

(4.51)

Shock at the right boundary: (a) $\rho ^\star \ne \rho _0$: From (4.17) and (4.19) one gets

$$\begin{aligned} H {| \, {\mu ^{L}}\, \rangle } & = (\tilde{h}^{-}_{1} + h_{L-1,L} + \tilde{h}^{+}_{L}) {| \, {\mu ^{L}}\, \rangle } \nonumber \\ & = \epsilon ^{\ell }(\rho _0) {| \, {\mu ^{L}}\, \rangle } + {| \, {\rho _0}\, \rangle }^{\otimes (L-2)} \otimes [(h + \tilde{h}^{+}_2) {| \, {\rho _0}\, \rangle } \otimes {| \, {\rho ^\star }\, \rangle }], \end{aligned}$$

(4.52)

$$\begin{aligned} H {| \, {\mu ^{L + 1}}\, \rangle } & = (\tilde{h}^{-}_{1} + \tilde{h}^{+}_{L}) {| \, {\mu ^{L + 1}}\, \rangle } \nonumber \\ & = \epsilon ^{\ell }(\rho _0) {| \, {\mu ^{L}}\, \rangle } + {| \, {\rho _0}\, \rangle }^{\otimes (L-1)} \otimes (h {| \, {\rho _0}\, \rangle }). \end{aligned}$$

(4.53)

Computations analogous to the treatment of the left boundary lead to

$$\begin{aligned} H {| \, {\mu ^{L}}\, \rangle } & = \left( d^{\ell } + d_+(\rho ^\star ,\rho _0)\right) {| \, {\mu ^{L}}\, \rangle } \nonumber \\{} & {} - d^{\ell }{| \, {\mu ^{L-1}}\, \rangle } + d_+(\rho ^\star ,\rho _0) {| \, {\mu ^{L+1}}\, \rangle }, \end{aligned}$$

(4.54)

$$\begin{aligned} H {| \, {\mu ^{L+1}}\, \rangle } & = \left( d(\rho _0,\rho ^\star )\nonumber \right. \\{} & {} \left. + d_+(\rho _0,\rho ^\star )\right) \left( {| \, {\mu ^{L+1}}\, \rangle } - {| \, {\mu ^{L}}\, \rangle } \right) , \end{aligned}$$

(4.55)

and we arrive at

$$\begin{aligned} G_{yL}(\rho ^\star ) & = \left( d^{\ell } + d^{r}_{+}(\rho ^\star )\right) \delta _{y,L} - d^{\ell } \delta _{y,L-1} - d^{r}_{+}(\rho ^\star ) \delta _{y,L+1}, \end{aligned}$$

(4.56)

$$\begin{aligned} G_{yL+1}(\rho ^\star ) & = d^{\ell }_{+}(\rho ^\star ) \left( \delta _{y,L+1} - \delta _{y,L}\right) , \end{aligned}$$

(4.57)

where

$$\begin{aligned} d^{r}_{+}(\rho ^\star ) & = d_+(\rho ^\star ,\rho _0), \end{aligned}$$

(4.58)

$$\begin{aligned} d^{\ell }_{+}(\rho ^\star ) & = d(\rho _0,\rho ^\star ) + d_+(\rho _0,\rho ^\star ), \end{aligned}$$

(4.59)

(b) $\rho ^\star =\rho _0$: In the same fashion one gets from (4.18) by taking ${| \, {b}\, \rangle }={| \, {\rho _1}\, \rangle }$

$$\begin{aligned} H {| \, {\mu ^{L}}\, \rangle } = \left( d(\rho _0,\rho _1) + d_+(\rho _0,\rho _1)\right) \left( {| \, {\mu ^{L}}\, \rangle } - {| \, {\mu ^{L-1}}\, \rangle } \right) , \end{aligned}$$

(4.60)

so that

$$\begin{aligned} G_{Ly}(\rho ^\star ) = \tilde{d}^{\ell }_{+} \left( \delta _{y,L} - \delta _{y,L-1}\right) , \end{aligned}$$

(4.61)

where

$$\begin{aligned} \tilde{d}^{\ell }_{+} = d^{\ell } + d_+(\rho _0,\rho _1). \end{aligned}$$

(4.62)

To summarize, with $\chi$ defined in (3.11) and $\lambda$ defined in (3.17), the boundary rates appearing in G are given as functions of the shock density $\rho ^\star$ by:

(i) $\rho ^\star \notin \{\rho _0,\rho _1\}$

$$\begin{aligned} d^{\ell }_{-}(\rho ^\star ) & = d^{\ell } - \omega _- \frac{\chi }{1-\chi } \, = \, d^{\ell } + \lambda \omega _-, \end{aligned}$$

(4.63)

$$\begin{aligned} d^{r}_{-}(\rho ^\star ) & = (d^{r} + \omega _-) (1-\chi )^{-1} \, = \, (d^{r} + \omega _-) (1-\lambda ), \end{aligned}$$

(4.64)

$$\begin{aligned} d^{\ell }_{+}(\rho ^\star ) & = (d^{\ell } + \omega _+) \chi ^{-1} \, = \, (d^{\ell } + \omega _+) \frac{\lambda }{\lambda -1}, \end{aligned}$$

(4.65)

$$\begin{aligned} d^{r}_{+}(\rho ^\star ) & = d^{r} - \omega _+ \frac{1-\chi }{\chi } \, = \, d^{r} + \frac{1}{\lambda }\omega _+. \end{aligned}$$

(4.66)

(ii) $\rho ^\star = \rho _0$

$$\begin{aligned} d^{\ell }_{-}(\rho _0) & = d^{\ell }, \end{aligned}$$

(4.67)

$$\begin{aligned} d^{r}_{-}(\rho _0) & = d^{r} + \omega _-, \end{aligned}$$

(4.68)

$$\begin{aligned} \tilde{d}^{\ell }_+ & = d^{\ell } + \omega _+. \end{aligned}$$

(4.69)

(iii) $\rho ^\star = \rho _1$

$$\begin{aligned} \tilde{d}^{r}_{-} & = d^{r} + \omega _- , \end{aligned}$$

(4.70)

$$\begin{aligned} d^{\ell }_{+}(\rho _1) & = d^{\ell } + \omega _+, \end{aligned}$$

(4.71)

$$\begin{aligned} d^{r}_{+}(\rho _1) & = d^{r}. \end{aligned}$$

(4.72)

The last step is to prove positivity of the rates. In terms of the boundary rates of the ASEP, the boundary rates of the random walk $\rho ^\star$ are given by

$$\begin{aligned} d^{\ell }_{-}(\rho ^\star ) & = \frac{\alpha (1-\rho ^\star ) - \gamma \rho ^\star }{\rho _1 -\rho ^\star } \, = \, \frac{\alpha (1-\rho ^\star )}{z_0} \frac{z_0-\kappa _-(\alpha ,\gamma ) z^\star }{\rho _1 -\rho ^\star }, \end{aligned}$$

(4.73)

$$\begin{aligned} d^{r}_{-}(\rho ^\star ) & = \frac{(r - \ell ) \rho _1(1-\rho _{1}) - \alpha (1-\rho _{1}) + \gamma \rho _{1}}{\rho _1 - \rho ^\star }, \end{aligned}$$

(4.74)

$$\begin{aligned} d^{\ell }_{+}(\rho ^\star ) & = \frac{(r - \ell ) \rho _0(1-\rho _{0}) + \delta (1-\rho _{0}) - \beta \rho _{0}}{\rho ^\star - \rho _{0}}, \end{aligned}$$

(4.75)

$$\begin{aligned} d^{r}_{+}(\rho ^\star ) & = \frac{\beta \rho ^\star - \delta (1-\rho ^\star )}{\rho ^\star -\rho _{0}} \, = \, \frac{\beta \rho ^\star }{z^\star } \frac{z^\star - \kappa _-(\beta ,\delta ) z_1 (1-\rho ^\star )}{\rho ^\star -\rho _{0}}, \end{aligned}$$

(4.76)

for any shock density. For shock density $\rho ^\star \in \{\rho _0,\rho _1\}$ one has

$$\begin{aligned} \tilde{d}^{\ell }_+ & = d^{\ell }_+(\rho _1) = d^{\ell } + \frac{\delta (1-\rho _{0}) - \beta \rho _{0}}{\rho _1 - \rho _{0}}\nonumber \\ & = (r-\ell )\rho _0 + \frac{\delta }{1-\rho _0}, \end{aligned}$$

(4.77)

$$\begin{aligned} \tilde{d}^{r}_{-} & = d^{r}_-(\rho _0) = d^{r} - \frac{ \alpha (1-\rho _{1}) - \gamma \rho _{1}}{\rho _1 - \rho _{0}} \nonumber \\ & = (r-\ell )(1-\rho _1) + \frac{\gamma }{1-\rho _1} . \end{aligned}$$

(4.78)

The jump rates $d^{r,\ell }$ do not depend on the shock density and are positive by construction. The boundary jump rates (4.77) and (4.78) are manifestly positive, and since $0\le \chi \le 1$ one concludes that the rates $d^{\ell }_+(\rho ^\star )$, and $d^{r}_-(\rho ^\star )$ as given in (4.64) and (4.654.66) are positive for all shock densities $0\le \rho ^\star \le 1$. Since the function $\kappa _-(\cdot ,\cdot )$ is strictly negative for strictly positive rates, one finds that $d^{\ell }_-$ as given in (4.734.744.75) is negative for $\rho ^\star > \rho _1$ and that $d^{r}_+$ as given in (4.76) is negative for $\rho ^\star < \rho _0$. Thus the shock density has to be chosen such that $\rho _0 \le \rho ^\star \le \rho _1$ to avoid negative boundary rates for the random walk.

Thus all cases envisaged in the theorem are covered and in the $(L+2)\times (L+2)$-matrix

$$\begin{aligned} G(\rho ^\star ) = \left( \begin{array}{ccccccccc} d^{r}_{-} &\quad - d^{\ell }_{-} &\quad 0 &\quad 0 &\quad \dots &\quad 0 &\quad 0 &\quad 0 &\quad 0 \\ -d^{r}_{-} &\quad d^{\ell }_{-} + d^{r} &{} -d^{\ell } &{} 0 &{} \dots &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} - d^{r} &{} d^{r}+d^{\ell } &{} -d^{\ell } &{} \dots &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} - d^{r} &{} d^{r}+d^{\ell } &{} \dots &{} 0 &{} 0 &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} d^{r}+d^{\ell } &{} -d^{\ell } &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} - d^{r} &{} d^{r}+d^{\ell } &{} - d^{\ell } &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} 0 &{} - d^{r} &{} d^{\ell } + d^{r}_{+}&{} - d^{\ell }_{+} \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} 0 &{} 0 &{} - d^{r}_{+} &{} d^{\ell }_{+} \end{array}\right) , \end{aligned}$$

(4.79)

for $\rho _0< \rho ^\star < \rho _1$ one recognizes the form of the generator of a biased random walk with nearest-neighbor jumps and reflecting boundaries.

The random walk dynamics for $\rho ^\star =\rho _0$ and $\rho ^\star =\rho _1$ are the same except for a shift of the lattice by one site and generated by the $(L+1)\times (L+1)$-matrix

$$\begin{aligned} G = \left( \begin{array}{cccccccc} \tilde{d}^{r}_{-} &{} - d^{\ell } &{} 0 &{} 0 &{} \dots &{} 0 &{} 0 &{} 0 \\ -\tilde{d}^{r}_{-} &{} d^{\ell } + d^{r} &{} -d^{\ell } &{} 0 &{} \dots &{} 0 &{} 0 &{} 0 \\ 0 &{} - d^{r} &{} d^{r}+d^{\ell } &{} -d^{\ell } &{} \dots &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} - d^{r} &{} d^{r}+d^{\ell } &{} \dots &{} 0 &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} d^{r}+d^{\ell } &{} - d^{\ell } &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} - d^{r} &{} d^{\ell } + d^{r} &{} - \tilde{d}^{\ell }_{+} \\ 0 &{} 0 &{} 0 &{} 0 &{} \dots &{} 0 &{} - d^{r} &{} \tilde{d}^{\ell }_{+} \end{array}\right) . \end{aligned}$$

(4.80)

For vanishing boundary barriers $\omega _+=\omega _-=0$ this random walk is homogeneous, i.e., the boundary jump rates are the same as the bulk jump rates. For $\rho ^\star \notin [\rho _0,\rho _1]$, the intertwining relation (2.6) with dual matrix $G(\rho ^\star )$ holds, but one jump rate is negative.

4.3.2 Dual random walk with absorbing boundaries

In the following, we take $\sigma _0=1$ and $\sigma _1=-1$, prove the existence of a matrix $G^-$ such that $H {| \, {\theta ^x}\, \rangle } = \sum _{y} G^-_{yx} {| \, {\theta ^x}\, \rangle }$, and show that $G^-$ is the generator of a random walk with absorbing boundaries.

The bulk equation (4.34) does not depend on the choice of $\sigma _0$ and $\sigma _1$ so that

$$\begin{aligned} G^-_{yx} = G_{yx}, \quad 2\le x \le L-1. \end{aligned}$$

(4.81)

The treatment of the boundary terms is also analogous, but there is one crucial difference. For the boundary term involving $\tilde{h}^{-}$ we use (4.18) and observe that for $a_0 = 1$ and $a_1 = -1$ one has $d_-(\rho ^\star ,b) = 0$. Therefore

$$\begin{aligned} \tilde{h}^{-} {| \, {\sigma \rho _1}\, \rangle } = \left( \tilde{d}_-(\sigma ,b) - \tilde{d}(\sigma ,b)\right) {| \, {\sigma \rho _1}\, \rangle } + d(\sigma ,b) {| \, {b\rho _1}\, \rangle }, \end{aligned}$$

(4.82)

with an arbitrary vector ${| \, {b}\, \rangle }$ linearly independent of ${| \, {\rho ^\star }\, \rangle }$. Choosing ${| \, {\tau _3}\, \rangle } = {| \, {b}\, \rangle } = {| \, {\rho _1}\, \rangle }$ as in the general case gives $\tilde{d}_-(\rho ^\star ,\rho _1) = \alpha +\gamma$ and thus leads to the projection property

$$\begin{aligned} (\tilde{h}^{-}_1 + h) {| \, {\rho ^\star \rho _{1}}\, \rangle } & = \left( r + \alpha +\gamma \right) {| \, {\rho ^\star \rho _{1}}\, \rangle } - d^{r} {| \, {\rho _{0}\rho ^\star }\, \rangle } . \end{aligned}$$

(4.83)

From (3.37) one then gets

$$\begin{aligned} H {| \, {\theta ^{1}}\, \rangle } & = \left( d^{r} + \alpha +\gamma \right) {| \, {\theta ^{1}}\, \rangle } - d^{r} {| \, {\theta ^{2}}\, \rangle }, \end{aligned}$$

(4.84)

and we arrive at

$$\begin{aligned} G^-_{y1} = \left( d^{r} + \alpha +\gamma \right) \delta _{y1} - d^{r} \delta _{y2} ,\end{aligned}$$

(4.85)

instead of (4.46) for the general case. The difference measure $\theta ^{0} = \mu ^{0}(0) - \mu ^{0}(1)$ vanishes identically so that $G_{y0}=0$ for all y.

Similarly, one obtains for the right boundary

$$\begin{aligned} H {| \, {\theta ^{L}}\, \rangle } & = \left( d^{\ell } + \beta + \delta \right) {| \, {\theta ^{L}}\, \rangle } - d^{\ell } {| \, {\theta ^{L-1}}\, \rangle }, \end{aligned}$$

(4.86)

and therefore

$$\begin{aligned} G^-_{yL}(\sigma ) = \left( d^{\ell } + \beta + \delta \right) \delta _{yL} - d^{\ell } \delta _{y,L-1}, \end{aligned}$$

(4.87)

instead of (4.564.57) for the general case. Also $\theta ^{L+1} = 0$, which yields $G^-_{yL+1}=0$ for all y.

Hence the set of vectors ${| \, {\theta ^{x}}\, \rangle }$ with $1\le x \le L$ spans an L-dimensional subspace of $({\mathbb C}^2)^{\otimes L}$ that is invariant under the action of H. The $L\times L$-matrix

$$\begin{aligned} G^- = \left( \begin{array}{ccccccc} d^{r} + \alpha +\gamma &{} - d^{\ell } &{} 0 &{} \dots &{} 0 &{} 0 &{} 0 \\ -d^{r} &{} d^{\ell } + d^{r} &{} -d^{\ell } &{} \dots &{} 0 &{} 0 &{} 0 \\ 0 &{} - d^{r} &{} d^{r}+d^{\ell } &{} \dots &{} 0 &{} 0 &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \dots &{} d^{r}+d^{\ell } &{} - d^{\ell } &{} 0 \\ 0 &{} 0 &{} 0 &{} \dots &{} 0 &{} d^{\ell } + d^{r} &{} - d^{\ell } \\ 0 &{} 0 &{} 0 &{} \dots &{} 0 &{} - d^{r} &{} d^{\ell }+ \beta + \delta \end{array}\right) , \end{aligned}$$

(4.88)

is the generator of a biased random walk with absorbing boundary conditions with absorption rate $\alpha +\gamma$ at the left boundary site 1 and absorption rate $\beta + \delta$ at the right boundary site L.

4.3.3 Random walk duality

We recall that the shock measures with density $\sigma ^\star$ can be expressed in terms of shock measures with a density $\rho _0$ by the invertible transformation matrix M (3.20). This implies that generally shock measures for one shock density $\rho ^\star$ can be expressed by a linear transformation $M(\rho ^\star ,\sigma ^\star )$ in terms of a linear combination of shock measures with a different shock density $\rho ^\star$. This implies the similarity transformation

$$\begin{aligned} G(\sigma ^\star ) = M^{-1}(\rho ^\star ,\sigma ^\star ) G(\rho ^\star ) M(\rho ^\star ,\sigma ^\star ) ,\end{aligned}$$

(4.89)

between the two different random walks with reflecting boundaries on a lattice of $L+2$ sites as long as neither of the densities is equal to $\rho _0$ or $\rho _1$. Since the random walk is reversible also duality and reverse duality follow.

To discuss the singular case $\rho ^\star \in [\rho _0,\rho _1]$, we note that the projection $P M G(\rho ^\star ) M^{-1} P$ of the transformed generator for generic shock densities on the $(L+1)$-dimensional subspace spanned by the vectors ${| \, {\mu ^x(\rho _0)}\, \rangle }$ yields the generator $G(\rho _0)$. This yields the projective intertwining duality

$$\begin{aligned}{}[G(\rho _0) P M - P M G(\rho ^\star )] P = 0, \end{aligned}$$

(4.90)

with the duality matrix PM. A similar duality for $G(\rho _0)$ arises from projection on the $(L+1)$-dimensional subspace spanned by the vectors ${| \, {\mu ^x(\rho _1)}\, \rangle }$.

Also the difference between reflecting and absorbing boundaries for the shock random walk is just a difference of mathematical description of the random walk problem, not a difference in its physical properties, even though it at first sight looks physical. This is an example of Siegmund duality [76], which can be illustrated in a slightly simplified setting as follows: Consider a symmetric random walk with jump rates 1 and with reflecting boundaries at site 0 and L + 1. The probability distribution satisfies the master equation $\dot{P}(k,t) = P(k+1,t) + P(k-1,t) - 2 P(k,t)$ for $2\le k \le L-1$ and the boundary equations $\dot{P}(1,t) = P(2,t) - P(1,t)$, $\dot{P}(L-1,t) = P(L,t) - P(L-1,t)$. Now consider the difference of probabilities defined by $Q(k,t) := P(k+1,t) - P(k,t)$. For this quantity, the master equation yields the evolution equation $\dot{Q}(k,t) = Q(k+1,t) + Q(k-1,t) - 2 Q(k,t)$ for $2\le k \le L-2$ for the bulk and $\dot{Q}(1,t) = Q(2,t) - 2 Q(1,t)$, $\dot{Q}(L-1,t) = Q(L-2,t) - 2Q(L-1,t)$. This is the master equation of a random walk with absorbing boundaries at sites 0 and L. For the shock random walk, the absorbing boundaries arise from generic reflecting boundaries by considering the evolution of the difference measure $\theta ^k$, which is analogous to considering the probability difference Q(k, t) in the illustration above.

4.3.4 A note on the spectrum

Spectral properties of the generator H of the open ASEP have been studied numerically [52, 53] using the density matrix renormalization group for non-Hermitian quantum Hamiltonians [79] and analytically via the Bethe ansatz [54, 78, 80,81,82,83,84]. The main difficulty in the Bethe approach is the construction of a reference state for excitations, which is not straightforward due to the lack of particle number conservation at the boundaries. Duality provides a different approach by diagonalizing the generator G of the dual process, which is conservative and hence can be studied with a standard coordinate Bethe ansatz. Eigenvectors of G then yield immediately a subset of eigenvectors of H on the parameter manifold defined by 2.10 in a basis spanned by the probability vectors given by the shock measures.

The generator G is a tridiagonal Toeplitz matrix, which makes the computation of the spectrum and eigenvectors a well-known textbook problem. The eigenvectors are superpositions of plane waves and reflected plane waves on the lattice. For general boundary barriers with $\omega _\pm \ne 0$, the wave numbers are noninteger and cannot be represented in closed form except (i) for the stationary distribution (3.38) with eigenvalue 0 and (ii) for $L-1$ further eigenvectors when the boundary parameters are on the special submanifold (2.13) considered in [25]. With (3.4), this manifold can be expressed by the constraint $\omega _+ \omega _- = r \ell$ or, using also (3.36) and (3.37),

$$\begin{aligned} \omega _+ \omega _- = r \ell = d^r d^\ell , \end{aligned}$$

(4.91)

which rules out vanishing boundary barriers. The $L-1$ linearly independent eigenvectors computed in [25] have integer wave numbers like in the case when only condition (2.10) with $N=1$ is satisfied but $\omega _+ = \omega _- = 0$.

However, the generator G is of dimension $L+1$, which implies that one linearly independent eigenvector is missing for the submanifold (2.13). To find this eigenvector, we consider the absorbing duality and observe that, according to (3.36) and (3.37), the absorption rates can be represented in the form $\alpha +\gamma = d^\ell + \omega _-$ and $\beta +\delta = d^r + \omega _+$. Thus any eigenvector with components $f_u(x)$ and eigenvalue $\varepsilon (u)$ satisfies the recursion

$$\begin{aligned} - \varepsilon f_u(x) = d^\ell f_u(x+1) + d^r f_u(x-1) - (d^\ell + d^r) f_u(x) , \quad 2 \le x \le L-1, \end{aligned}$$

(4.92)

together with the boundary condition

$$\begin{aligned} - \varepsilon f_u(1) & = d^\ell f_u(2) - ( d^r + d^\ell + \omega _-) f_u(1), \end{aligned}$$

(4.93)

$$\begin{aligned} - \varepsilon f_u(L) & = d^r f_u(L-1) - (d^\ell + d^r + \omega _+) f_u(L). \end{aligned}$$

(4.94)

The ansatz $f_u(x) = a u^x$ yields the three equations

$$\begin{aligned} - \varepsilon & = d^r (u^{-1}-1) + d^\ell (u - 1), \end{aligned}$$

(4.95)

$$\begin{aligned} - \varepsilon & = d^\ell u - ( d^r + d^\ell + \omega _- ) ,\end{aligned}$$

(4.96)

$$\begin{aligned} - \varepsilon & = d^r u^{-1} - (d^\ell + d^r + \omega _+). \end{aligned}$$

(4.97)

Adding the second and third equation to the first equation yields

$$\begin{aligned} \varepsilon = \alpha + \beta + \gamma + \delta , \end{aligned}$$

(4.98)

so that the remaining two equations reduce to

$$\begin{aligned} 0 & = d^\ell u + \omega _+, \end{aligned}$$

(4.99)

$$\begin{aligned} 0 & = d^r u^{-1} + \omega _-. \end{aligned}$$

(4.100)

With the constraint (4.91), these equations are equivalent so that an eigenvector with

$$\begin{aligned} u = - \frac{d^r}{\omega _-}, \end{aligned}$$

(4.101)

and eigenvalue (4.98) exists. It is noteworthy that this eigenvalue appears also in the symmetric simple exclusion process with open boundaries on the manifold (2.13). For this case there exists a two-dimensional time-dependent representation of the dynamical matrix algebra [7, 85] for time-dependent matrix product measures as is the case for diffusion-limited pair annihilation with pair creation [86].

5 Conclusions and some open problems

The main result is an intertwining duality between the open many-body ASEP and a simple one-particle biased random walk that holds on the special manifold (2.10) of the boundary rates with $N=1$. We identify a one-parameter family of duality functions that are the probabilities of Bernoulli shock measures with a single shock with an arbitrary shock density $\rho ^\star$ as well as a special duality function that is given by the difference of the Bernoulli shock measures with a single shock and shock densities 0 and 1, respectively. This allows for the explicit computation of the full time evolution of the open ASEP if the initial distribution belongs to this family of shock measures and the duality asserts that the shock position performs a random walk. In contrast to related work [25] where dualities with shock measures involving many shocks were obtained for the further constraint (2.12) with $k=1$, we consider here only duality functions corresponding to a single shock, but without restriction to the further constraint on the boundary rates. Computing eigenvectors for the random walk problem with particle number conservation then yields eigenvectors for the nonconservative open ASEP on the parameter manifold (2.10) with $N=1$ in a basis spanned by the probability vectors given by the shock measures.

Depending on the choice of the duality function, the random walk has absorbing or reflecting boundaries. We compute the bulk and boundary transition rates of the random walk as a function of the parameters of the ASEP which was left open in [57], where only a dual reflecting random walk with shock density $\rho _0$ on a lattice of $L+1$ sites was conjectured. The general reflecting case corresponds to a random walk of a shock with density $\rho ^\star$ on a lattice with one or two auxiliary boundary sites. The reversibility of the walk allows for reformulating the intertwining duality as a reverse duality, which in turn can be expressed as a conventional duality for the time-reversed ASEP. The absorbing case, which corresponds to the special duality function, is reminiscent of the absorbing duality of the symmetric simple exclusion process [12, 22, 75].

The stationary distribution of the ASEP on the parameter manifold considered in this work is known to be given by a two-dimensional matrix product measure [40] with representation matrices that have a lower-triangular band structure. As shown in [57], this particular representation is a convex combination of shock measures with shock density $\rho _0$. Here we find that triangular similarity transformations of the matrix representation induce linear transformations between shock measures with different shock densities. Consequently, the different dualities analyzed here in detail can be related by a similarity transformation given in explicit form in (3.20).

At first sight, the dependence of the boundary conditions for the random walk on the duality function may appear surprising: On the one hand, one would generally expect the spectrum of the generator of a random walk to depend on the boundary conditions. On the other hand, by duality, the spectrum in the present case is a subset of the spectrum of the generator of the open ASEP and therefore cannot depend on the choice of duality function and hence not on the boundary conditions of the random walk. The transformation property between shock measures resolves this apparent contradiction as it shows that the different boundary conditions are related by similarity transformations that leave the spectrum of the generator invariant and that also imply a duality between random walks with different boundary conditions generated by a similarity transformation. This can be understood as a Siegmund duality between reflecting and absorbing boundaries for random walks [76].

We note that also ASEPs with other update dynamics defined in discrete time allow for MPMs with two-dimensional representations [59, 87, 88]. Therefore the stationary distributions of these models are convex combinations of shock measures. Random walk dynamics of the shock has been proved only for the degenerate model with deterministic bulk dynamics discussed in [59]. It would be interesting to search for a reverse duality in these models. A further interesting open problem is the existence of two-dimensional stationary matrix product states for the $U_q[\mathfrak {sl}(2)]$-symmetric ASEP with partial exclusion [89] and in more general reaction–diffusion processes with exclusion [1, 90,91,92]. If such representations exist, one may go on and search for reverse dualities that yield random walk dynamics of domain walls in such classical stochastic many-body systems. Similarly, one may investigate whether the four-dimensional representation of the MPM for the coagulation–decoagulation model [93] can be expressed in terms of shock measures. For reaction–diffusion systems, the shock would correspond to a reaction front.

Many other questions remain open, and we mention some just for the usual ASEP discussed in this work. The first is a generalization to duality functions for N shocks for boundary conditions (2.10) for the ASEP that admit $(N+1)$-dimensional representations of the matrix product measure [40]. A particular problem here is to understand how a change of representation changes the form of the shock measures and under which conditions absorbing rather than reflecting dualities arise. One then also expects to gain insight in the submanifold (2.12) of boundary parameters where the $(N+1)$-dimensional representations exist only for large enough system size $L\ge k$, which has been investigated in some detail in [62] where a link to Askey–Wilson polynomials is derived but where no connection to shock measures is mentioned. It is intriguing that Askey–Wilson polynomials are related to Meixner polynomials, which play an important role in duality theory [23, 30] and appear also in the representation theory of the quantum algebras $U_q[\mathfrak {sl}(2)]$ [94, 95] and $U_q[\mathfrak {su}(1,1)]$ [96].

The next step could then be the derivation of reverse or intertwining duality for the completely general case of boundary rates of the open ASEP where only infinite-dimensional representations of the matrix algebra exist. Also a study of time-dependent finite-dimensional representations of the dynamical matrix product algebra for the ASEP would be interesting.

Finally, we remark that the duality function (4.26) for shock density $\rho ^\star =1$ is proportional to the single-particle self-duality function obtained in [28] for the conservative time-reversed ASEP with reflecting boundaries corresponding to $\alpha =\beta =\gamma =\delta =0$. This self-duality is a consequence of the symmetry of the generator under the quantum algebra $U_q[\mathfrak {sl}(2)]$ [94, 97] that, however, is broken for open boundaries [98]. It would be interesting to understand why this duality function appears also in the open system on the manifold (2.10) discussed in this work for $N=1$ and whether this property extends to general N. It is tempting to conjecture that the dual process will be the shock exclusion process with color exchange [18] that arises from the self-duality of the conservative priority ASEP with more than one species of particles and $U_q[\mathfrak {sl}(N+1)]$ symmetry that remains integrable also for open boundaries [99, 100].