Lifted TASEP: long-time dynamics,generalizations, and continuum limit

We are grateful to the four referees for their comments on our manuscript. In the following we reply to the various points they have raised. The changes to our manuscript are indicated in colour.

Report 1:

We thank the referee for their report. The referee states that

"The minor revision that I suggest is optional: Maybe the authors could add a brief discussion of whether or not the observed $1/L^2$ scaling of the spectral gap at the critical point is related to a domain wall random walk. As proved rigorously in a recent paper [J. Phys. A: Math. Theor. 56 274001 (2023)], such a collective random walk phenomenon gives rise to a $1/L^2$ scaling of the spectral gap in the usual asymmetric simple exclusion process with open boundaries. Such domain walls appear to exist also in period exclusion processes with a defect particle, as studied in Ref. [18] and in depth recently in a noteworthy paper on arXiv [https://arxiv.org/abs/2411.08480]. Looking into this might provide more physical insight into the findings in the present work on the lifted TASEP."

We thank the referee for pointing out these references and their interesting suggestion that there might be a connection of these works to the $1/L^2$ scaling of the gap we observe at $\alpha_{\rm crit}$. At the moment, we unfortunately have no insights with regards to a potential connection of our finding to domain-wall motion, and we therefore don't feel comfortable to add a discussion on this possibility.

Report 2:

We thank the referee for their report and helpful comments and suggestions. Our responses to the various points the referee makes are as follows:

1. It's unclear why the 'breakdown' is unique to $\alpha_{\rm crit}$. Please clarify what changes for other $\alpha$.

As we have shown in the manuscript, the discrepancy between Bethe Ansatz results and Monte-Carlo simulations occurs only at $\alpha_{\rm crit}$. For other values of $\alpha$ the scaling of the spectral gap obtained from Bethe Ansatz agrees with the relaxation time (extracted from the autocorrelation times of the structure factor and other observables) obtained by Monte-Carlo. The physics for $\alpha \neq \alpha_{\rm crit}$ (KPZ scaling) is very different from one at $\alpha_{\rm crit}$, which is determined by the true self-avoiding random walk. On a more technical level, we stress that changing $\alpha$ leads to very complex rearrangements of the eigenvalue spectrum of the transition matrix, and different classes (in terms of the Bethe Ansatz equations) of eigenstates give the gap-scaling in the three different regions of $\alpha$. In light of this it is arguably not surprising that the "breakdown" occurs only at $\alpha_{\rm crit}$.

1. No practical proposal is provided for an alternative way to measure the relaxation time that would actually confirm the Bethe-ansatz analysis.

To the best of our understanding there simply isn't a numerical way of obtaining the "true" relaxation time by considering the dynamics close to equilibrium. Our understanding is summarized in section 3.4. This leaves numerical methods for determining the spectral gap of the transition matrix. Given that the particle number must be large (in order to deal with the level-crossings at large N seen in the Bethe ansatz) the only approach that might be capable of doing this would be based on matrix-product state methods. We are not expert in these, which is why we have not attempted such an analysis.

We have added a comment stating that it would be interesting to carry out an analysis of the spectral gap of the transition matrix for large N using MPS methods.

1. At the end of the paper, a generalization of the lifted TASEP is presented. However, the motivation behind this definition is not well laid out, and the results are very preliminary, which is not sufficient to fully appreciate the potential of this generalization. A simple question that could have been addressed is the following: Does this generalization correspond to a lifted version of any known model in the literature? Additionally, the abstract states, By tuning the pullback parameter in the GL-TASEP to a particular value we can again achieve a polynomial speedup in the time required to converge to the steady state.'' I did not find a follow-up to this statement in the main text. Please clarify this point.

We thank the referee for this remark and for the associated question. The generalized lifted TASEP (which is indeed a lifing of a generalization of the TASEP) shows that we can construct models of particles hopping on a one-dimensional lattice that resemble the lifted TASEP in that only a single particle can move, that have a critical alpha, and that feature $N^{5/2}$ scaling of the autocorrelation times away from \alpha_crit and $N^{3/2}$ scaling at $\alpha = $\alpha_{\rm crit}$. All choices of $\pi_1 <= \pi_2 <= \pi_3$ ... yield consistent models, and we expect all of them to have critical alphas with $N^{3/2}$ scaling, a conjecture that we have explicitly checked for the model with $\pi_1 = p$. In the present manuscript, we are particularly interested in whether the simplest generalization, the one with $\pi_1 = p$, is integrable. The criterion of Sá, Ribeiro and Prosen suggests integrability, but we cannot, at present, confirm this property. We agree with the referee that this complex matter was presented much too briefly.

In the revised manuscript, we discuss all the aforementioned points in a long paragraph that we have added after eq. (109).

1. Regarding the presentation of the article, I find that it can be improved for easier reading: a- The definition of the model at the top of page 2 is not clear enough: "a single particle being active. It carries a pointer which allows it to move in a forward direction or to undergo a collision. In the second part of the move, the pointer itself moves to the nearest neighbor to the left, which becomes the new active particle.'' This description does not fully match the figure, which can be confusing at first glance. In addition, It's note clearly stated that the model is built on a discrete-time TASEP, since the usual TASEP is in continuous time.

We thank the referee for these comments. We have amended the discussion of the model.

1. Figure 1 is a bit ambiguous. First, I assume the small horizontal dashes represent the simulation results; then what do the continuous lines represent? Are they just connecting the dashes? While one can accept that the values α =0.34 and 0.5 indeed follow the predicted behavior, the other values do not have anything to compare.

We thank the referee for these remarks. We have changed the figure, as follows: Dashed lines are for the asymptotes, $\sim N^{5/2}$ and $\sim N^{3/2}$. Simulation points are given by dots, and the full lines connect them. This is now explained in the caption, which also mentions that the $\sim N^{5/2}$ scaling for $\alpha \neq \alpha_{\rm crit}$ can only be seen for large N for $\alpha \lesssim \alpha_{\rm crit}$. In order to illustrate this last point, we have added a second $\sim N^{5/2}$ asymptote that follows the data for large N for $\alpha = 0.46$. Just below eq. (8), we already briefly alluded to this point. We extended the discussion there.

1. Page 6 paragraphs are repetitive… condense or reorganize. Additionally, raw numerical data in the text do not offer particular insights, in particular on page 13.

We thank the referee for these comments. However, we believe that the manuscript would become less readable by making the suggested changes. On p.6 we introduce the three states that we then investigate in detail. This is done by providing the necessary information for readers to be able to reproduce our work, and stressing crucial differences (like the total momentum). There is some repetition as the same information in provided for all three states, but this is unavoidable. The raw numerical data the referee refers to on page 13 contains very valuable information from a Bethe Ansatz point of view: it gives the roots, which uniquely identify the eigenstate. We included the explicit root distributions because the way we "follow" a given Bethe state in L is highly non-standard, and we wanted to ensure that the manuscript contains sufficient information for interested readers to reproduce our analysis.

1. More minor issues: - Equation (10) seems to me valid only for N odd; otherwise, the solution zα=1,E=1 does not hold. Please clarify.

We thank the referee for this comment. Equation (10) holds for N even as well as for N odd. When $E=1$ and all $z_a$ tend to 1, the factors in the product on the right-hand-side become ill-defined (for both N even and N odd). It is therefore convenient to consider the equivalent equations where both sides are multiplied by $(E-\alpha-(1-\alpha)/z_a)^N$. It is then clear that $z_a=1$ is a solution with $E=1$.

1. It is not clear where the fitting ansatz in Equation~(17) comes from. Is it purely empirical ?

Yes, it is a simple empirical fit, which however reproduces the data very convincingly.

1. There is a typo on page 12: the susceptibility that at best behaves as''

We thank the referee for spotting this -- we have fixed it.

1. I am a bit confused by the notation: at the end of the paragraph Factorized Metropolis filter,'' it defines p=p1. This suggests you do not need generic pk anymore, yet they are used in the following paragraph. Please clarify.

We thank the referee for this remark. Clearly, our description of the algorithm was too sketchy. What we meant to say is that the generalized lifted TASEP can be defined for any choice of ${\pi_0, \pi_1, \pi_2...}$ as long as we have that $\pi_0 <= \pi_1 <= \pi_2$ .... The case where $\pi_0=0, \pi_1 = 1$ etc.

In response to the referee's remark, we have extended the discussion in the header of Section 5.1, which now presents much more detail. We now discuss what the choices of the Boltzmann factors $\pi_0 <= \pi_1 <= \pi_2...$ imply for the Metropolis acceptance probabilities $p_0, p_1, p_2$, ..., and we state in detail that the Lifted TASEP corresponds to $p_0, p_1, p_2... = 0,1,1,...$ (with an explanation in words), and that the simplest variant of the generalized Lifted TASEP corresponds to $p_0, p_1, p_2... = 0,p,1,1....$ Finally, we mention explicitly that for all choices of $\pi_0 <= \pi_1 <= \pi_2 ....$, the stationary state is given by the Boltzmann distribution of Eq. (104).

We think that this revised version is now much clearer.

Report 3:

We thank the referee for their report. We completely agree that it would be great to have an observable that exhibits the scaling predicted by the Bethe ansatz. We have indeed spent a large amount of time and effort to find such an observable, but have failed to do so. We believe that this is for the reasons set out in section 3.4. See also our reply to Report 2 above.

With regards to the referee's doubt whether the kind of problem we encounter is specific to the LTASEP or is more general our view is as follows. We are not aware of any general principle that would rule out the scenario we propose in section 3.4 (which is consistent with all of our results) and we therefore think the problem we encounter is not specific to the LTASEP, but occurs more generally. However, as the example of the LTASEP shows, it is rather difficult to see that there is a problem in the first place! If one examines the LTASEP by standard numerical techniques like exact diagonalization of the transition matrix and MCMC one will not see that the extracted scaling of the relaxation time with system size disagrees with the spectral gap for large particle numbers. We hope that rigorous work in mathematics will eventually fully clarify the complex scaling behavior of the lifted TASEP and its generalizations.

Report 4:

We thank the referee for their careful reading of our manuscript and very helpful comments, to which we now reply in detail.

1. The analogy with a quantum Hamiltonian is standard for the generator of continuous-time Markov chains. In the discrete-time case, however, the transition matrix corresponds to the exponential of minus the "Hamiltonian". This initially caused some confusion for me, as the spectral gap acts as an energy scale, and the eigenvalue of the transition matrix might be better denoted by a symbol like λ, rather than E, to avoid misleading associations.

Our notations follow our original paper, and we would like to keep them for that reason. In order to avoid confusion, we have added a sentence stressing that we are dealing with a discrete time process and that the eigenvalues are hence dimensionless.

1. In Eqs. (17), (29), and (31), I believe the left-hand side should be −Re(ln(E(L))) rather than E(L).

We thank the referee for spotting these typos! We have corrected them in the revised manuscript.

1. Regarding the plots of −Re(lnE), I suspect that the horizontal axis represents a quantity like 1/L raised to a certain power $L^{-5/2}$ in Figs. 1–4, 7, and 8; $L^{-3/2}$ in Fig. 5; and $L^{-2}$ in Fig. 6.

These are all log-log plots, and hence the axis label $1/L$ is the correct one. In order to make this clear we have included a statement that these are log-log plots in the relevant figures.

Additionally, in Fig. 5, the label $L^2$ should be corrected to $L^{-2}$.

We thank the referee for spotting this typo, which we have corrected in the revised manuscript.

1. The definition of $L^{(1)}_{\rm co}$ (first appearing on page 6, before Eq. (28)) is unclear. Could the authors clarify this?

$L^{(1)}_{\rm co}$ is an $\alpha$-dependent length scale that is defined by the asymptotic eigenvalue scaling providing an accurate description of the gap for system sizes $L>L^{(1)}_{\rm co}$. We have added a comment and formula explaining this when we first introduce this length scale.

1. To improve readability, especially in power series expansions such as in Eqs. (17) and (29), I suggest labelling the coefficients by the corresponding power.

We agree with the referee that these are superior notations. However, changing them would require us to redo all figures -- resulting in a significant amount of work. We think that the notations we use, while admittedly inferior, are not misleading and allow the reader to extract all necessary informations. We therefore would prefer to keep them.

1. Page 6: The reasoning behind the selected α-ranges for states 1, 2,and 3 is not clear. Why is α≤αcrit=1/2 considered only for state 1, \alpha > \alpha_{\text{crit}} = 1/2 only for state 2, and α=αcrit=1/2 only for state 3?

We thank the referee for this question. The reason is that in the different $\alpha$-regimes different states (in terms of the Bethe ansatz equations) give rise to the eigenvalues with the smallest real parts. We have considered a large range of states in the various $\alpha$-regimes and eventually focused on the ones that give the smallest gaps for large $L$.

We have added an explanation of this in the text.

1. Page 6, before Eq. (23): There is a stray "1" in the line containing P=2πL.

We thank the referee for spotting this issue. The "1" corresponds to a footnote, that the SciPost styles did not display. We have included the footnote in brackets instead.

1. Section 2.3.5: It is stated that "The imaginary part of the eigenvalue vanishes", which appears to contradict Eq. (32), where the eigenvalue for state 3 is complex.

We thank the referee for spotting this! The imaginary part is indeed non-zero. We have corrected this.

1. Page 15, before Eq. (72): Please correct the typographical error "number number".

We have corrected this typo.