Universality class of the mode-locked glassy random laser

1- The authors have performed extensive computer simulations of the Mode-Locked 4-phasor model beyond the narrow-bandwidth approximation.

2- The authors have implemented a delicate finite size scaling (FSS) analysis, and the procedure is clearly explained.

3- The authors have introduced a new set of boundary conditions for this problem, to ease their FSS analysis.

1- The justification of the model used by the authors is not totally well explained.

2- The arguments sustaining the mean-field scaling laws proposed by the authors are not elaborated enough.

The paper of Niedda et coworkers studies the Mode-Locked 4-phasor model beyond the narrow-bandwidth approximation, for which mean-field theory cannot be computed exactly, because the model is no longer fully-connected. They thus resort to extensive simulations on GPUs in order to address the statistical properties of the model. In particular, they show that its thermodynamic properties are similar to the ones of mean-field glassy models, for which a one-step replica symmetry breaking occurs at a static transition temperature.

In order to demonstrate this important result, the authors introduce new boundary conditions to have a better control on finite-size effects. This allows them to perform an extensive finite size scaling analysis of the heat capacity of the model for various system sizes, from which critical exponents can be extracted to compare with mean-field theory. They eventually discuss the replica symmetry breaking at the lasing threshold and determine the Parisi order parameter for glassy systems, namely, the overlap probability distribution among samples for the same realization of the disorder (which is eventually averaged over the disorder).

The paper is rich and overall well-written. However, I found hard to understand some of the arguments regarding the scaling laws. As a result, I recommend publication in SciPost Physics if the following points are taken into account by the authors.

- As previously said, the main problem I encountered while refereeing is about the scaling laws provided by the authors. They claim that the critical exponents for a standard second order critical point (and also for the REM) are $\nu=2$ and $\alpha=2/3$. However, I naively thought that they should be $\nu=1/2$ and $\alpha=0$ (as for standard scalar $\phi^4$-theory). To justify their claim, the authors use a scaling argument that I find hard to understand. In particular, they approximate the fluctuations below the critical point by the distance of the minimum of the Landau free energy from the origin, while the curvature of the Landau free energy at the minimum should be considered instead.

- I do not understand the last paragraph of the left column page 3, where it is written that the Fourier transform of $a_k$ should be roughly a Dirac delta function at $\omega_k$. Indeed, $a_k$ should be a slowly-varying amplitude, and thus its Fourier transform should only display low frequencies centered around 0, and not around $\omega_k$. I guess the authors discuss the Fourier transform of the electric field rather than the one of the slow amplitude $a_k(t)$, but maybe I am missing a point here.

- Second paragraph of page 4, I do not understand the end of the sentence ", and in the case of cavityless systems also compensate the leakages". I suspect one word is missing or replaced by another one.

- Just above Section III, the authors say that "As already mentioned (and without loss of generality) we will consider the J's as real parameters". I could not find to which part of the paper the authors were referring. Does it mean that the authors assume to be in the purely dissipative regime?

- At the beginning of the third paragraph page 5, the authors simplify the problem in the limit $\gamma\ll \delta\omega$, and conclude that $\mathcal{H}_2$ is an additive constant to the Hamiltonian. I do not understand why, since the couplings are a priori different from one mode to another, and the spherical constraint cannot be factorised out.

- Is there a reason why Eq. (19) is not just written $|k_a-k_b| = N - |k_a-k_b| if |k_a-k_b| \gtrsim N/2$? Besides $N/2$ is not well defined for odd $N$, and so the integer part should be considered.

- In Appendix B, I do not understand the argument for the choice of the scaling function [below Eq. (B3)]. Could you explain more?

- At the end of Section IV, the authors conclude that the model belongs to a mean-field universality class because the value of $1/\nu$ is consistent with this statement. Actually, the only rigorous statement that can be made is that the universality class of the model is not incompatible with a mean-field one. How do the authors conclude that the universality class is indeed a mean-field one? I think I am missing a point here.

- Could you justify why the complexity for one step goes from $N^2$ to $\ln N$ via parallelisation on GPUs, as claimed in the Conclusion?

- In the Conclusion, the authors claim that the RSB occurs at the lasing threshold. But from the data of Fig. 11, it seems that the the secondary peaks in the overlap distribution appear at much lower temperature (about 0.4) than the critical temperature $T_c$ (about 0.6). Is it expected that the two temperatures coincide in the large $N$ limit?

In addition, I noticed few typos in addition to the ones already noted by the other referee, that I list below:

- Last paragraph of page 2, there is a typo in "heterogeneous".

- Last paragraph of page 3, there is a typo, it should be "responsible for the Kerr effect".

- Beginning of the second column page 4, there is a typo in "heterogeneous".

- Just after Eq. (2), $\sigma_p^2\sim N^{p-2}$ should be replaced by $\sigma_p^2\sim N^{2-p}$.

- There is a problem of reference in the footnote [61].

- At the end of the first column page 8, there is an "in" missing between "presented" and "following subsection".

- One should change the notation for the exponent $x$ below Eq. (B4) since $x$ is already used for something else above.

- If I understand correctly, the coefficient $C$ in Eq. (B3) should be negative (so that the heat capacity is maximum at the transition). One should thus add absolute values in the very last equation of Appendix B because $\ln C_N$ and $\ln \tilde C_N$ are not properly defined.

- In the caption of Fig. 9, "differ from each other" should be "differs from the others".

1- Elaborate the scaling arguments.

2- Clearly indicate the assumptions behind the model used eventually for the simulations (only a quartic coupling, and real parameters).

3- Provide the simulation details for the REM (at least in an Appendix).

4- Discuss where the error bars are coming from in Figs. 6, 7 and 8 (at least in the Appendix).

5- Clarify the Appendix B about the finite size scaling analysis protocol.

6- Clarify the argument to conclude that the model is of mean-field nature.

7- Correct the various typos.

1- The authors present solid numerical results for the mode-locked 4-phasor model and effectively combine them with analytical predictions based on mean-field scaling arguments.

2- The model and the motivations for studying it are extensively discussed in the Introduction, which is very well-written and gives a complete presentation of the subject.

3- The authors exploit modified boundary conditions (PBC instead of FBC) for the frequencies to reduce the finite-size effects dramatically and obtain more accurate predictions of critical exponents.

The authors present strong evidence that the critical properties of the mode-locked 4-phasor model, characterized by the Frequency Matching Condition (FMC), can be described by a mean-field theory. This is achieved by combining extensive numerical simulations, analytical arguments based on the scaling hypothesis to control the finite-size effects, and the introduction of periodic boundary conditions on the frequencies to reduce finite-size effects in the numerics.
The problem addressed in the paper is notoriously very difficult and relevant both from a theoretical and experimental point of view. The techniques and results discussed represent a step forward in the comprehension of the model considered.
On this basis, I think the paper fully meets the Journal's criteria. I, therefore, suggest publication once some minor issues will be addressed by the authors.

1- Just above Eq. (10), the authors introduce the ratio $\mathcal{P}$ between the pump rate and the spontaneous emission rate. According to what the authors say, the spontaneous emission rate is proportional to the temperature of the bath $T$, while the pump rate is proportional to the energy of the system $\mathcal{E}=\epsilon N$. Consequently, I would expect $\mathcal{P} \propto \epsilon/T$. However, Eq. (10) states that $1/\mathcal{P}^2 = T/\epsilon^2$, which seems to be inconsistent with the definition. Can the authors clarify this point?

2- In Sec. V the authors present numerical results for the overlap distribution across the glass transition. They say they used $\mathcal{N}$ equilibrium configurations of replicas to accumulate statistics, but only in Appendix A the authors relate the quantity $\mathcal{N}$ to the Monte Carlo steps of the algorithm and no reference to the equation in the Appendix is present in the main text. I would suggest adding a reference to the Appendix so that the reader can have a better understanding of the meaning of $\mathcal{N}$. Moreover, neither in the caption of Fig. (9) nor of Fig. (10) there is an indication of the typical value of $\mathcal{N}$ in the numerical results presented. Is it possible to add it?

3- In Eq. (2) the notation $\sum_{\kappa|\mathrm{FMC}(\kappa)}$ is not immediately clear as there is no reference to Eq. (4), where the meaning of $\mathrm{FMC}(\kappa)$ is presented. I suggest adding a reference to Eq. (4) below Eq. (2) to ease the reading.

4- In order to facilitate the understanding of the plots, I would add a "$T$" beside the color maps in Figs. (1), (3), (4), (5), (10), and (11), so that it is immediately clear what different colors stand for.

5- There are some typos throughout the paper.
In the abstract "using , with" doesn't seem to be correct.
At the beginning of page 2, in the sentence starting with "In standard mode-locked lasers ..." there is a parenthesis that is opened but not closed.
Just before Eq. (5) "dymamics" should be replaced with "dynamics".