Random Matrices and the Free Energy of Ising-Like Models with Disorder

1 - Derivations very clear and easy to follow
2 - Paper is generally well written
3 - Topic is interesting and timely

1 - The construction of the model seems a bit 'artificial' at places
2 - Sometimes the mathematical derivations conceal or cloud the physical interpretation (or at least sometimes it would be desirable to better weave physical considerations on the Ising model with surface disorder into the mathematical analysis)

I enjoyed reading this paper, which is exceedingly well written and easy to follow. I have a few minor comments and requests for clarifications.

1 - Pag. 3 'elude' should probably be 'allude'
2 - An explicit expression for the matrix Q defined in Eq. 3 is never provided. It is true that it is essentially proven irrelevant in the following (at least as long as its eigenvalues are exactly on the unit circle), but it is somehow unsatisfactory that only its features (real skew-circulant etc) are provided, but not its explicit expression
3 - As the paper is written to be as self-contained and pedagogical as possible, I would cite more recent pedagogical introductions to RMT alongside Ref. [11] on pag. 4, for instance Potters' 'A first course in Random Matrix Theory' and Livan's et al. 'Introduction to Random Matrices: theory and practice'
4 - Section 2.3 would benefit from an explicit example (say, a 6x6 matrix) to show the structure of a general skew-circulant matrix. At the moment, it is difficult to picture in one's mind how such matrices look like from the definition in Eq. 12
5 - In Section 3, the reason why this precise choice of the disorder (skew-circulant) is made is unclear. Perhaps one should add that this choice simplifies (or makes it at all possible) a sounder analytical treatment, and briefly explain why it is so. Also, it is not completely clear to me what this choice actually means in terms of 'surface spins' - if I were to simulate this model on a computer, how should the spins $\pm 1$ on the outer interface be drawn to precisely reproduce this skew-circulant model?
6 - After eq. 54 'deriviated' (typo) ---> differentiated
7 - The first sentence on pag. 12 sounds quite odd to me. Why should it be 'remarkable' that the first cumulant is an incomplete Gamma function? Is this very expected or highly unexpected for some reason? Also, how does this exact cumulant compare with the first cumulant of a log-normal distribution, which (I believe) is claimed to provide a good approximation for the full distribution of the free energy?
8 - Before eq. 64, it is again not clear to me on what grounds the coupling strength is scaled in this way with the system size. Since much of the rest of the paper is related to the mathematical analysis of different cases depending on $\alpha$, it would seem slightly unsatisfactory to leave the reader wonder whether this is simply an ad hoc choice, or if instead there are deeper physical reasons for assuming this scaling.
9 - All the remaining discussion is mathematically sound and interesting, but I would suggest tying in the mathematical results a bit better with some physical insight on the Ising physics. One has the feeling at places that the authors have been slightly 'carried away' in their desire to show how much could be done analytically on this model, at the expenses of a clearer connection to the physical model they ought to describe.
10 - In the Conclusions, there are two sentences very close to each other, where first the distribution is stated to tend to a Gaussian, and immediately afterwards it is stated to be log-normal. A quick reader may find this slightly contradictory. I would urge the authors to specify a bit better which cases are considered and under which limits, yielding to Gaussian *or* lognormal (but clearly not both simultaneously).

See above

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1 - extensive and detailed derivations
2 - clear motivation

1 - figure layout not optimal in some places

This work provides a clear and systematic approach to studying the emergence of log-normal distributions of the free energy for large disordered Ising systems and their asymptotic Gaussian form. Overall, the arguments provided are plausible and can be followed well. The manuscript is well organized, starting with a clear motivation, defining a specific problem, and solving it step-by-step. The high level of detail allows one to follow the derivations, ask critical questions (see below), and enable future work to reproduce (not done here) and build upon the present results.

For my report, I attempted to closely follow the derivations (and apologize for the extra time this took) to formulate the following comments that the authors may want to consider.

* I think the abstract should reflect more on the content of the paper. Large efforts are invested in the analytical derivations towards the characteristic function that is later "merely" numerically evaluated for the finite-size corrections to the scaling arguments for the limit of M->infty. I was initially under the impression that the work involved numerical simulations after reading "We show numerically [that] [...] and in the limit of infinitely large matrices the distributions become Gaussian".

* In Eq. (1) I fail to see why an additional variable $K_{ij}$ was introduced

* Given the high level of detail and its later importance, I was hoping for more details on the origin of $Q$ and $\kappa$.

* Below Eq. (23) there is a distinction between $(0)_0$ and $(0)_n$ but both are equal to 1, such that it could also be valid for $n\geq 0$.

* Below Eq. (46) it is noted that all eigenvalues of Q being on a unit circle is a particularly interesting case, but I thought this is by construction. Since the final part of Sec. 3 is so fundamental, I could imagine that making this crystal clear may be beneficial. Also, it is picked up in the last paragraph again.

* Eq. (59) seems odd because the first equation is identical up to the indices $\xi$ and $\xi +1$.

* p. 15 :The statement "logarithmic growth [...] with additional linear contributions" irritates me. It rather looks like linear growth (M/2) with logarithmic contributions (MlogM) but maybe I am missing sth.

* Fig. 1: which $\alpha$ is shown here? If this is tested numerically, it should be verified for different values of $\alpha$, especially since Eq. (74) is super explicit about the $\alpha$ dependence.

* Same question regarding the choice of $\alpha$ applies to the following figures.

* p. 16: When it is "plausible that Eq. 75 also holds for j>4" then the statement "Thus, the leading order of all higher order cumulants is always..." appears a bit strong. Maybe the word "implies that" would work better

* Below Eq.(79) it is stated that the equation is independent of $\alpha$ but it explicitly involves an $\alpha$. Of course, the limit being zero becomes independent of $\alpha$ but the exponential decay does depend on $\alpha$.

* Eq. (81) should be only valid for $j>1$

* p. 19 in the synopsis, I do not understand the statement that the leading order behavior is not dependent on $\alpha$ if there is a logarithmic behavior that depends on $\alpha$ for negative values. However, as stated above I believe the logarithmic behavior is a correction, or not?

* Fig. 5: One cannot see the x labels at all. As a suggestion, it would be much more comprehensible for me if the absolute relative error was plotted on a log scale. The same applies to Fig. 6

* p. 21 "Importantly, the plots show that the qualitative behavior is captured ..." -> I would argue that small relative errors imply that also the quantitative behavior is captured, or am I mistaking sth?

* p. 22: It was unclear to me at this point where "the numerical data" really came from.

* Table 1: I would suggest to start with the leading-order term. Otherwise, I was confused about the sign change for $\alpha=-1/4$ or the statement that "For all $\alpha>0$ the leading order of $\sigma$ will no longer be proportional to $M^{-1/2}$"

* Fig. 7a: Is this really $\kappa_1$ (figure axis) or $\kappa_3$ (caption)?

* Fig. 8/9: Why is there no crossover between log-normal and Gaussian approximation in sight? For large $M$ the Gaussian approximation should become much better, right?

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