Liu-Nagel phase diagrams in infinite dimension

1. The results presented in this paper are very interesting and will be potentially very important to understand the long reported anomalous behavior of amorphous materials and to rationalize the experimental response of disordered solids to externa stresses.
2. This article presents a detailed and simple description of the exact derivation of the Liu-Nagel diagrams in infinite dimensions for harmonic soft spheres, enabling to reproduce the results presented by the same authors in a recent paper [Nature Physics 12, 1130–1133 (2016)], something that would be extremely difficult without this technical paper.
3. Still, the results discussed here go beyond those presented in the previous paper, and extend to soft potentials results obtained recently for hard spheres in infinite dimensions, specially for the part concerning shearing and yielding.

1. The figures concerning the phase diagrams (mostly fig 1, the Nagel-Liu diagram) are difficult to interpret. The axes are not the traditional ones, and I find hard to abstract their meaning.
2. Although the first part of the paper is easy to follow, the part concerning the fullRSB results (from section V) is completely obscure and hard to follow. There is a constant reference to previous papers for derivations, equations become lists of equations where the variables that appear are not even defined...

As I indicated above in the Strengths section, I believe the paper is very interesting and will strongly contribute to the condense matter community, and for this reason, I cannot do other but recommend its publication om SciPost. However, I think that revisions are compulsory to make this paper more available to the community. I list below the requested changes.

General comments,

1. Fig 1 is hard to understand. The position of the liquid, supercooled liquid and the glass should be written in the figure. A cross-section surface of the figure, where the dynamical line becomes just a point, might help to understand the different transitions. Like it is now, I don't see a Gardner transition under cooling or compression.

2. I understand that the fullRSB derivations might be too long to include them in this paper, and considering that they are very similar to those presented in previous papers of hard spheres, the decision was to omit them and refer the previous papers. The problem is that, as they appear now, it's impossible to understand anything, I think it is only accessible to people familiar with the infinite dimensional hard-sphere calculations, which corresponds to a very reduced part of the community. I suppose the list of formulas concerning magnitudes not defined in the text aim to help a future reader trying to repeat the results, but in such a case, they would rather gather them in an appendix, as they are now they only contribute to confusion.

Now, minor comments,

3. From the beginning, the authors refer to the scaled packing fraction and temperature, unless I am wrong, I don't see the definition of this scaling anywhere.
4. Below Eq. (1), volume is defined as V, I think authors wanted to define N, the number of particles instead, I don't see V anywhere.
5. $s$ is not defined
6. I find a bit confusing the introduction of m, authors refer to the 'biased partition function' whose meaning is not discussed. Is it $\beta\to m \beta$?
7. The 1 in Eq. (13) shouldn't be a m?
8. I don't understand why an $Y$ does not appear in Eq. (33).
9. I find the reasoning below Eq. (39) very unclear.
10. The term $e^{h}$ that appears after the change of variable I suppose that arrives after taking the $d\to\infty$ limit, something should be mentioned.
11. At the end of Eq. (41) the a->c for the product.
12. What is (1) and (2) in Eq. (44).
13. I find hard to understand the justification for equations (45) and (46).