qZero Range has Random Walking Shocks
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Abstract
...but no other surprises in the zero range world. We check all nearest neighbour 1dimensional asymmetric zero range processes for random walking product shock measures as demonstrated already for a few cases in the literature. We find the totally asymmetric version of the celebrated qzero range process as the only new example besides an already known model of doubly infinite occupation numbers and exponentially increasing jump rates. We also examine the interaction of shocks, which appears somewhat more involved for qzero range than in the already known cases.
Keywords
Interacting particle systems Second class particle Shock measure Exact solution qZero range processMathematics Subject Classification
60K35 82C231 Introduction
Since the early investigations of their hydrodynamic scaling limits (e.g., Rezakhanlou [27]), it has been clear that several 1dimensional asymmetric interacting particle systems develop shocks; moving discontinuities of the rescaled density profile. Such shocks are very well understood on the hydrodynamic PDE level, but less so in the stochastic model itself. This microscopic phenomenon has been investigated by several authors, we give a brief summary of some of the important steps in the area.
Second class particles are coupling objects on the level of particles that trace the difference between two coupled instances of a particle model with slightly differing initial conditions. We describe the precise coupled dynamics for our models in Sect. 2. Briefly, second class particles use up the difference of jump rates between the coupled pair of models. Their connections to shocks have a long history of exploration, we only mention a few instances of this here. Derrida et al. [15] investigated stationary distributions in the asymmetric simple exclusion process (ASEP) as seen by a second class particle, and realised that their description greatly simplifies under certain condition on the jump of the densities in the shock. Namely, the stationary distribution becomes a product of Bernoulli distributions with different constant densities on the two sides of the second class particle. Balázs [3] reproduced this result in the exponential bricklayers process (EBLP). Next, Belitsky and Schütz [10] were able to transfer the result in a fixed frame of reference by observing that under the same condition on shock densities, the ASEP evolves a shock product distribution into translations of this same distribution, moreover the coefficients are the transition probabilities of a drifted simple random walk. We therefore call this phenomenon random walking shocks, and will give its precise formulation in Sect. 3.
Belitsky and Schütz [10] could also handle multiple shocks, exploring the interaction between them. Random walking shocks of ASEP interact like another ASEP, except the jump rates of these shockwalkers change with the rank as we consider them from left to right. This results in an attraction between the shockrandom walkers, making them appear as one large shock after rescaling. Balázs [4] again reproduced this result in the EBLP and noted that, rather than excluding each other, in this model the random walkers almost act independently except for their rankbased jump rates. This again makes the walkers attract each other and a group of microscopic shocks appear as one large shock in the hydrodynamic scale. A precise statement for multiple random walking shocks follows in Sect. 3.
The random walking shock results had not used the second class particle until Balázs et al. [7] reproduced both random walking shock results (for ASEP and EBLP) with a second class particle added to the position where the densities jump. This on one hand simplified the arguments, on the other hand explained both the stationarity results from the viewpoint of the second class particle, and the random walking shock measures that first came without the second class particle. They also added a generalised exponential zero range process; a doubly infinite extension of zero range processes with an exponential form of the jump rates. This process came naturally as it has half the dynamics of EBLP. Certain other processes (e.g., [7, 24]) are known to possess the random walking shock property, to the best knowledge of the authors none of which belong to the zero range family.
In this work we perform a systematic search within 1dimensional nearest neighbour attractive zero range processes and find the totally asymmetric qzero range (qTAZRP) as the only new model that has the random walking shocks property. This model is remarkable for its algebraic (and now, probabilistic) properties. We do not use dualities, however in light of recent developments our results give strong indications regarding the nature of dualities behind qTAZRP. Indeed, some details already emerged in private communications with Frank Redig—further investigations will be part of future work.
We believe that our result and purely probabilistic approach will also nicely complement the more analytic research that has been done on qTAZRP, some of which we briefly mention next.
Some of the early mentions of the model is by Povolotsky [23], where it is found as one suitable for Bethe ansatz. Remarkably, qTAZRP is one of the examples that satisfy the microscopic concavity property, hence Kardar–Parisi–Zhang (KPZ)scaling of current fluctuations could be proved using purely probabilistic methods in [8]. It also arises as the interparticle distance process for qTASEP, distortion of totally asymmetric simple exclusion and a very celebrated particle system in the integrable probability literature. Duality and determinantal formulas allow exact calculations in Borodin et al. [14] that show connections of qTASEP to the celebrated KPZ equation. It has also been shown that qTASEP arises as certain projection of McDonald processes, see Borodin and Corwin [13]. Bethe ansatz and integrable probability techniques allowed Korhonen and Lee [19] to derive exact formulas for certain transition probabilities in qTAZRP. Ferrari and Vető [16] and Barraquand [9] proved TracyWidom limits for current fluctuations in qTASEP, while recently Imamura and Sasamoto [18] proved KPZrelated scaling limits of the tagged particle distribution. Finally, we mention Barraquand [9] and Lee, Wang [21] as investigations for the nonhomogeneous version of the dynamics.
In this article we stick to simple probabilistic methods, to be more precise we apply the arguments of [7] to carry out our search. For this reason at several points we simply refer to [7] rather than repeat all arguments from therein. Our notation fully conforms [7]. We start by recalling parts of the zero range framework in Sect. 2, then state and prove our results in Sects. 3 and 4.
2 Zero Range Process

the order \(\omega _i(t)\le \zeta _i(t)\) is a.s. kept for any time \(t\ge 0\),

marginally \(\underline{\omega }(t)\) follows the zero range dynamics,

marginally \(\underline{\zeta }(t)\) also follows the zero range dynamics.
3 Random Walking Shocks
Here is our first result of a systematic search within the class of zero range processes above for random walking shocks.
Theorem 1
 1.
\(I={\mathbb {Z}}^+\), \(g(y)={\mathfrak {a}}y\), \({\mathfrak {a}}>0\), \(\theta =\sigma \), \({\hat{\mu }}=\mu ^\theta =\mu ^\sigma \); then \(P=p{\mathfrak {a}}\) and \(Q=(1p){\mathfrak {a}}\).
 2.
\(I={\mathbb {Z}}^+\), \(g(y)={\mathfrak {a}}\cdot \bigl (1{\mathrm{e}}^{\beta y}\bigr )\), \(p=1\), \(\beta ,{\mathfrak {a}}>0\) and \(\theta +\beta =\sigma <\ln {\mathfrak {a}}\), \({\hat{\mu }}=\mu ^\sigma \); then \(P=\bigl (1{\mathrm{e}}^{\beta }\bigr )\cdot \bigl ({\mathfrak {a}}{\mathrm{e}}^{\sigma }\bigr )\) and \(Q=0\).
 3.
\(I={\mathbb {Z}}^+\), \(g(y)={\mathfrak {a}}\cdot \bigl ({\mathrm{e}}^{\beta y}1\bigr )\), \(p=1\), \(\beta ,{\mathfrak {a}}>0\) and \(\sigma =\theta \beta \), \({\hat{\mu }}=\mu ^\sigma \); then \(P=\bigl ({\mathrm{e}}^{\beta }1\bigr )\cdot \bigl ({\mathrm{e}}^{\sigma }+{\mathfrak {a}}\bigr )\) and \(Q=0\).
 4.
\(I={\mathbb {Z}}\), \(g(y)={\mathfrak {a}}\cdot {\mathrm{e}}^{\beta y}+{\mathfrak {b}}\), \(p=1\), \(\beta ,{\mathfrak {a}}>0\), \({\mathfrak {b}}\ge 0\) and \(\ln {\mathfrak {b}}<\sigma =\theta \beta \), \({\hat{\mu }}=\mu ^\sigma \); then \(P=\bigl ({\mathrm{e}}^{\beta }1\bigr )\cdot \bigl ({\mathrm{e}}^{\sigma }{\mathfrak {b}}\bigr )\) and \(Q=0\).
 1.
This is the case of independent walkers. A second class particle in this model is yet another independent particle in the system with no interactions whatsoever, and it indeed sees a flat \(\mu ^\theta ={\hat{\mu }}=\mu ^\sigma \) scenario stationary. No shocks occur in this description, \(P\) and \(Q\) are the constant rates of a single walker.
 2.
This is qTAZRP, its rate can be rewritten as (a constant multiple of) \(g(y)=1q^y\), and the shock condition is \(\theta \sigma =\ln q<0\). The jump rate is concave, hence so is the hydrodynamic flux [5]. A shock therefore jumps upwards in density as we cross it from left to right.
 3.
This is the convex counterpart of qTAZRP, only differs from the next case in having a lower bound on particle occupations. Shocks are negative jumps, accordingly.
 4.
This doubly infinite generalised exponential zero range has been known to have random walking shocks since [7], in fact its bricklayers counterpart, without the second class particles, was already investigated in [4]. A slight novelty here is the constant \({\mathfrak {b}}\) which has not been noted before in this context. Indeed, without the constant \({\mathfrak {b}}\) the marginals have the remarkable property that \(\mu ^\theta \) is a shift by one of \(\mu ^\sigma \). This property is lost with the introduction of \({\mathfrak {b}}\ne 0\) (it is obviously false in the above cases when the marginals are both concentrated on \(I={\mathbb {Z}}^+\)). Nevertheless, the random walking shocks structure survives this addition of \({\mathfrak {b}}\) in the rates.
We examine this phenomenon for cases 2, 3 and 4 from above.
Theorem 2
The interpretation is again that the product distribution (8) evolves into a linear combination of similar product measures with changed \(\underline{\sigma }\) and \(\underline{m}\) parameters, and the coefficients can be given an interacting random walks interpretation. To see this, notice that the steps \(\underline{m}^{i,i+1}\) and \(\underline{\sigma }^{i,i+1}\) keep (9) for all times. The walkers are the second class particles themselves, jumping from \(i\) to \(i+1\) with rate \(P(m_i,\,m_{i+1},\,\sigma _i,\,\sigma _{i+1})\). This is clearly a generalization of the random walking shock property (7) for a single shock.
The rates \(P\) give insights on the interaction between shocks. Notice first that the form we obtained for \(P\) is rather similar to the rates \(g\) of the model itself. This is something one has seen in [7] to some extent in both ASEP and EBLP, and it repeats here again. Multiple walkers are allowed to occupy the same site. In this case, we see stronger interactions than with EBLP before: except for \({\mathfrak {b}}=0\) in case 4, the rate \(P\) of jump from a site \(i\) will not coincide with the sum of the rates of individual shockwalkers. This is due to the convexity of the exponential function.
To see that shockwalkers stay in tight distance to each other, we can repeat the argument for Lemma 4.2 of [4]. Namely, first label the walkers in increasing order from left to right (and run the label dynamics to keep this order). Then notice that two neighbouring walkers, if both are alone on their sites \(i<j\), have higher rate to decrease their distance then to increase it. Adding higherlabeled walkers on site \(j\) blocks the front walker due to the labeling dynamics, while adding lowerlabeled ones on site \(i\) makes \(m_i\), hence the rate of decreasing the distance, even higher. Thus, each interwalker distance is stochastically bounded by independent Geometric distributions.
Thinking of hydrodynamics, it is natural to expect that our shockwalkers have the Rankine–Hugoniot velocity (4) as their mean velocity. We verify this directly in certain cases:
Proposition 3
The Rankine–Hugoniot formula (4) holds for the velocity in single shock cases of Theorem 1, and the twoshock cases of Theorem 2.
Due to the more complicated interaction of shocks, we do not see an easy way to verify this for more than two shocks, as has been done for the examples of [7].
There is a well established connection between exclusion and zero range processes that works via mapping interparticle distances of exclusion into zero range models. The classical ASEP maps to constant rate asymmetric zero range, and qTASEP maps to qTAZRP this way. Notice that random walking shocks in the form discussed here are not known to appear in constant rate zero range, nor in qTASEP. The mapping of course gives something, but this seems rather unnatural as the second class particles of the respective models do not get maped into each other either. The random walking shocks of ASEP, with [7] or without [10] the second class particle, give a strange statement for constant rate zero range that the authors can best imagine by simply undoing the mapping back to ASEP, which does not do much good to the state of the art. The random walking shocks discovered here for qTAZRP also mean something in qTASEP, but this also seems to be best imagined by undoing the mapping back to qTAZRP. In fact, it is via the mapping that already flat, constant density stationary distributions of qTASEP are best understood.
4 Proofs
We build on [7], and cite various steps and formulas from there across our proofs.
Proof of Theorem 1
If \(p<1\) then this implies in the first line that \(g\) is an affine function, and since it cannot go negative, the range of occupations cannot be doubly infinite, that is \(I={\mathbb {Z}}^+\). Furthermore, this line also tells us that \(\mu ^\theta ={\hat{\mu }}\), which in turn is equal to \(\mu ^\sigma \), and this gives case 1 of the theorem.
When \(I={\mathbb {Z}}^+\), \({\hat{{\mathfrak {a}}}}>0\) requires again \(\theta >\sigma \), and we need \({\mathfrak {b}}={\hat{{\mathfrak {a}}}}\) for \(g(0)=0\). This is case 3. However, we can also have \({\hat{{\mathfrak {a}}}}<0\) with \(\theta<\sigma <{\bar{\theta }}=\ln {\mathfrak {b}}\) and \({\mathfrak {b}}={\hat{{\mathfrak {a}}}}\). This is case 2 after a sign change in the constant, \({\mathfrak {a}}={\hat{{\mathfrak {a}}}}\).
Proof of Proposition 3 (single shock case)
Proof of Theorem 2
Proof of Proposition 3 (two shocks)
For case 3, change the sign of \({\mathfrak {a}}\), while for case 4, turn \({\mathfrak {a}}\) into \({\mathfrak {b}}\). In both cases change the sign of \(\beta \) as well. In the last step for case 3, substitute \({\mathfrak {b}}={\mathfrak {a}}\) as was done in that case. Otherwise the proof is word for word identical. \(\square \)
Notes
Acknowledgements
The authors thank anonymous referees for pointing out typos and ways to improve the manuscript. M.B. also thanks Frank Redig for valuable discussions on duality and its connections to random walking shocks. M.B. was partially supported by the UK Engineering and Physical Sciences Research Council Standard Grant EP/R021449/1.
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