Generalized hydrodynamics in complete box-ball system for $U_q(\widehat{sl}_n)$

We thank all the referees for their positive, detailed and constructive reports. We have taken their suggestions into account in the new version of the manuscript. We answer all the comments and suggestions just below as well as in the ‘List of changes‘ box.

[Referee #1] But on the science side, the one thing I am curious about is what is the correction to the diffusive behaviour seen in the numerics. It is mentioned that the diffusive GHD prediction is not quite reached by the numerics, but that the error decreases. How does it decrease? With a power law? Given how accurately GHD can be recovered, this would be an interesting thing to observe in order to guide future work on corrections beyond diffusive. Maybe the authors can make a quick remark about this?

The shape of the transition between two consecutive plateaus is analysed at the end of Sec 5.3. In Fig. 10 the results of the simulations are compared to the error functions that are predicted by diffusive corrections to GHD (Sec 5.2). The agreement is good in all cases, but, as pointed out by the referee, we observed one case where the agreement is quantitatively not as good (but it improves as time grows). As written at the end of Sec 5.3 the reason for such behaviour is not clear to us. The referee’s suggestion to carry a quantitative analysis of the convergence is certainly good. One reason why this has not yet been done is the fact that a reliable analysis would presumably require even more precise simulation results (increased number of initial conditions) while the numerical data presented in Fig. 10 already represent some rather significant amount of CPU time (~10^16 applications of combinatorial R and thus several months of CPU time).

[Referee #2] In Section 5.3, I can't see the difference between the different t plots (even on my 'retina' display). While I understand that this is the point - it seems like bad practice to graphically represent two sets of data in a way where you can't properly see either because of overlap with the other. I leave it to the authors to consider if they can solve this issue.

Presenting a large quantity of numerical data carrying a lot of information in a few plots can often be tricky. For the problem raised by the referee one option would be to keep a single value of t in each plot, but this would no longer show the readers that the data actually overlap, which is an important result. In addition we stress also that the curves do not overlap everywhere since they visibly differ in the steep regions of each step, and this illustrates the diffusive broadening. A common practice in this kind of situation is to shift vertically the curves associated to different values of t. But in the present case, because of the “thickness” of the signal (due to the fluctuations effect) this does not seem doable in practice (it would require very large shifts to separate the curves, and they would no longer fit inside each panel). For the above reasons we do not see any satisfactory way to improve the presentation of the data.

[Referee  #3] I think it is an awkward nomenclature to call cBBS ystem’s configurations as “states”, and then treatment of distributions over configurations - which in the more standard language of statistical mechanics would correspond to “states” - as “randomized cBBS”. Maybe the authors can rethink that, but my suggestion would be to use “configurations”, for the former, and “states” for the later, while the system (cBBS) has no intrinsic randomness (it is deterministic in both cases) so it does not make sense to call it “randomized cBBS” in case where statistical ensembles of cBBS configurations are considered.

In fact, state is used for single microscopic configurations everywhere except in Sec. 5, and we believe that the precise meaning is always quite clear from the context. Still, we have added a comment (footnote #12) at the beginning of Sec. 5 to warn the readers that, in this section, state should be understood in a thermodynamic (or probabilistic) sense. We have removed the expression “randomize cBBS” from the abstract and from the beginning of the introduction. It is then used for the first time in page 4, where we have added a sentence to stress that randomized refers to the initial conditions, and that the dynamics remains fully deterministic. We also note that “randomized BBS” has already been used several times in the literature.

> [Referee #1] the dot before “non diagonal”?
Corrected.

> [Referee #1] monotonous -> monotonic

Corrected.

> [Referee #2] (...) the figures are referred to variously as fig 1, Fig. 1, Figure 1. Please be consistent.

Done, thanks.

> [Referee #2] There seems to be a problem with the ordering of figures: the discussion of Figure 10 comes directly after the discussion of Figure 6. Please fix.

The (previously numbered) figure 10 has been moved after Fig. 6. So the old Fig. 10 is now Fig. 7.

> [Referee #2] In Section 6 it is stated that '[...] the simulations match perfectly the GHD expectations'. Either make this quantitative (with a percentage error) or don't state it.

Since the quality of the agreement between the simulations and the GHD theory is quite obvious in all figures, we have not carried out a quantitative estimate of the error. We have therefore removed “ match perfectly” and have replaced it with the weaker statement “are in good agreement with”.

> [Referee #2] Section 6 gives a nice summary of what is a fairly long and technical paper. It would help with the readability of the paper if the authors could cross-reference (i.e. include equation numbers for) the earlier results in the paper they are referring to in this concluding section.

Done.

> [Referee #2] I think there deserves to be some discussion of the role of the diffusive corrections presented in Section 5.2 in the comparison with the numerics.

The calculation of the diffusive corrections described in Sec. 5.2 is indeed directly relevant to the broadening of the steps between plateaux, as observed in the numerics and in Fig. 10 in particular. This is in fact already discussed in the paper. We have added one comment at the end of Sec. 5.2 to indicate that the error-function form (and the quantity \Sigma) is compared quantitatively to the simulation data at the end of Sec. 5.3.

> [Referee #3] Is commutative diagram (2.48) completely correct, namely the two vertical mappings T^{(k)}_l are not identical. One is a conjugation of the other by the map \Phi. See also last in-text formula on the page 16.

The referee is right, formally the two vertical mappings are different functions. However, we feel that it would be unnecessarily heavy to introduce here a new notation here, since the two mappings are naturally identified. We have added a footnote (called just above the diagram (2.48)) to indicate that the same name is used for two formally distinct objects.

> [Referee #3] Maybe notation on the RHS of (2.52) has to be explained?

We have added a sentence below (2.52) to explain that the graphical (box) notation is explained in the appendix C.2.

> [Referee #3] There were other integrable deterministic cellular automata in recent literature, where the interplay between ballistic and diffusive transport has been established, even beyond the hydrodynamic assumptions. See e.g. Commun. Math. Phys. 371, 651 (2019), or PRL 119, 110603 (2017). I don't expect that such explicit results could be achieved for cBBS system, but perhaps the authors can comment on this and/or make make an appropriate link to this relevant literature.

We thank the referee for drawing our attention to these two interesting references. They are now cited in the revised manuscript (in Sec. 6).