Universality class of the mode-locked glassy random laser

Dear editor, herewith we send our replies to the referees comments. We amended the manuscript accordingly. The most relevant changes are in red in the text.

In the following we answer point to point to the referees remark.

Best regards Luca Leuzzi, on behalf of all the authors.

Report 2

Referee: The authors have taken into account most of the comments I made about the first version of their work. This second version is far clearer. I have though still few minor remarks: To be consistent with the comment from my previous report, the sentence "The outcome of the scaling analysis is that the mode-locked random laser is in a mean-field universality class" in the abstract should be toned down, as the work of the authors only show that the data are compatible with a mean-field universality class.

Reply: we changed the text in the abstract as “The outcome of the scaling analysis is that the mode-locked random laser universality class is compatible with a mean-field one”.

Referee: In page 2, sparse networks are not introduced. At least, the scaling of the number of interactions with N should be indicated.

Reply: We added a footnote, [46], where we explain that “By sparse networks we mean that the average connectivity of each variable does not scale with the number $N$ of variable and, therefore, the total number of couplings in the systems grows like $N$.”

Referee: - In Eq. (36), ξ∞ should be to power d so that one recovers ω=ψ/ν

Reply: The referee is right, we corrected the formula as observed.

Referee: What A stands for in Eqs. (A2) and (A3)?

Reply: It should have been $a$, rather than $A$, and it is a mode configuration. We changed the lettercase and we added the symbol $\bm a_t$ after configurations.

Referee: The authors did not fully take into account my comment about the computation of error bars. In Appendix A, they mention that the error bars come from the disorder average, but I am wondering how they compute them in practice? Is it the standard deviation of the canonical averages for several realizations of the disorder, or is it a more sophisticated quantity (for instance obtained via the bootstrap or the jackknife method)?

Reply: We are sorry for that. Using jackknife (or bootstrap) did not change a lot the outcome, possibly because we can basically neglect the thermal fluctuations with respect to the quenched disorder fluctuations in computing the statistical error. We explain this in Appendix A in the revised version: “We observed that taking data uncorrelated in time, in view of the fact that the time average contribution to the error is negligible with respect to the quenched disorder contribution and using the numbers $N_s$ of simulated random samples indicated in table \ref{tab4}, the leading digit of the statistical error practically does not change when it is computed using anti-distorsion techniques such as jackknife and bootstrap with respect to a simple standard deviation computation on the sample-to-sample fluctuations. The errorbars in the figures are, therefore, all computed as standard deviations.

Referee: Several typos should also be corrected

Reply: We thank the referee for the careful reading. We have corrected all the typos indicated.

Report 1 Referee: In the second version of the manuscript, the authors have clarified the doubts I had about some points of their manuscript; in particular the points (1) -(4) in my report. I have noticed, however, that the typos identified at point (5) of my report have not been addressed, so I wonder if this is because the authors believe they are not typos or if it is just a forgetfulness. I report here the typos I identified.

Reply: We apologize for our forgetfulness. We have corrected all the typos indicated by the referee in the revised version.

All indicated typos corrected.

A footnote has been added to explain sparsity in the introduction.

Eq. (36) corrected.

In appendix A a paragraph has been added about the computation of the statistical error.