Joint distribution of currents in the symmetric exclusion process

1- Obtention of new results via innovative and advanced techniques.
2- The paper is well-written and the derivations detailed.

1- Numerical methods could be detailed.

See attached file.

1- subject of high interest currently
2- advanced techniques
3- new results that were previously difficult to get

1- The results are not compared with numerical simulation of SEP itself
2- It is not addressed how the integral equations are solved numerically, and if their solutions are indeed expected to exist and be unique
3- The derivation, especially concerning the initial condition and the appearance of delta functions, could be clearer

The paper provides exact results for the full, joint large-deviation theory for two quantities in SEP: the total current at a point $Q_t$, and the total current along a diffusive-shape curve $J_t$. This is a new result, of high interest in current research. It is based on the macroscopic fluctuation theory (MFT), and a mapping, recently discovered, of the MFT equations to the AKNS integrable PDEs. With this mapping, one can obtain exact solutions by using the inverse scattering method. The mapping is not new, but the application made here extends the applicability of the methods significantly. Many non-trivial steps are required, in particular the authors clarify the boundary conditions and their physical meaning. The results, relatively complicated but numerically workable integral equations, are very interesting. From the viewpoint of the Scipost criteria, I would say the present works help for “Open a new pathway in an existing or a new research direction, with clear potential for multipronged follow-up work”.

I believe the paper should be published in Scipost Physics, after the points below are addressed.

1- Perhaps the main criticism for this paper is the lack of a numerical check against SEP simulations. Figure 1 shows comparison between the numerical solutions of the integral equations, and a numerical solution of the MFT equations. However, it should be not too hard to simulate SEP itself, and verify at least some of the new results here. Of course, one expects MFT to work, however the results go beyond what has been done before, so this looks like a natural thing to do.

2- The integral equations (8,9) with (11-17) are relatively complicated, but it looks like they are indeed solvable numerically. But it looks as though the author have not addressed very much how they solved these equations (or how they solved the MFT equations). An appendix on this would be useful. Another related point: are the integral equations expected to have unique solution? Is a solution even expected to exist? This should be discussed.

3- In the part of the derivation in section 5.2.1, the mapping to AKNS is made. However, I find the discussion of the appearance of delta functions unclear. The mapping gives a precise expression for $u,v$ in terms of $q,p$, yet it appears to give ill-defined delta-function terms because of the choice of step initial condition, and instead of using the precise mapping, the authors just write arbitrary coefficients for the delta functions which they then determine by other means. In fact, even the normalisation of $u,v$ is changed, to make one coefficient 1, while these functions defined in a precise way. This is confusing. In principle, one could take a smoothing out of the initial condition, and there should be not problem; a limit towards the step should give the precise form of the delta function coefficient. Why is this not done? Is what is done equivalent? Why can one change the normalisation of $u,v$ (above eq 74) - is this properly taken into account later on when expressing results for the physical observables?

Some additional small comments:

Eqs 5,6: also depend on xi in principle

Eqs 20 and those that follow: what is rho?

Eq 23: these decay are independent of the initial densities rho_+, rho_-? Is there an understanding why this is so?

Eq 32: it would be useful to give the expressions of $D(r)$ and $\sigma(r)$

Paragraph below eq 48: $\hat \psi_\xi$

Eq 59: this is the same as eq 32, probably should be referenced here.

In sect 3, the initial density profile is $\rho_0(x)$, eventually specialised to the step initial condition. Make it more clear that you start more general, and when you specialise to the step initial condition. For instance, in discussing time reversal, around eqs 64-65, it seems that you take a general $\rho_0(x)$ but then in fact it looks like you need the special step condition because you use $\mu(\rho_+)$ and $\mu(\rho_-)$, while in fact after eq 67 you say that the resulting eqs are valid for any $\rho_0(x)$.

Before eq 74 (already mentioned above): it is true that AKNS has this invariance, but the functions $u, v$ were {\em defined} from $q,p$ in eq 69. So with this change of normalisation, it looks as though the relation of $u,v$ to $q,p$ is changed.

1- compare with SEP simiulations
2- clarify numerical methods used and solutions of integral equations
3- clarify derivation as per above.

1) This work presents an important advance of complete integrability in many body systems which will be of interest to specialists working in this field and on single-file motion.

2) The paper is well structured.

3) The paper presents detailed and clear derivations.

1) Some of the results could use a clearer physical interpretation at an earlier stage of the manuscript (see attached report)

2) The introduction might describe more clearly the hydrodynamics approach used here, how is it different from alternative microscopic approaches, and highlight the relation between the two (See attached report).

See attached report in pdf. format.

I have listed a number of suggestions in the attached report.