High-dimensional random landscapes: from typical to large deviations

I would like to thank both Referees for their careful reading of these lecture notes and for all their comments and suggestions.
In the v2 of the notes, their suggestions have been accounted for. Below, I give a list of the main changes that I have implemented.
Additionally, I am uploading a version of the v2 of the notes in which the main changes with respect to the v1 are highlighted in blue.

Main changes related to Referee Report 1:

• Corrected typos mentioned in points 1–4 and 6–10 of the referee's report:
• Removed the statement that the distribution has a “well-defined limit when $N \to \infty$”, replaced with “the scaled variable $\mathcal{N}(\epsilon)/N$ remains of order O(1) when $N \to \infty$”. Correction implemented in all other instances where the same phrasing appeared, including in Box [B5].

Main changes related to Referee Report 2:

• Included several comments to emphasize that the concepts and methods presented in the notes (with the aim of introducing tools for high-dimensional landscapes) represent a specific and not exhaustive approach to the inference problem. These comments are included in Sec. 1.2 when discussing the scope of the notes, in Sec. 2.3.1 when discussing optimality of maximum likelihood in the low-rank matrix estimation problem, in Sec. 2.3.3 when discussing gradient descent dynamics, and in Sections 3.3.1 and 3.3.3 when discussing the corresponding issues for low-rank tensor estimation. Clarified that the notes focus on continuous variables;
• Added Refs. [1]–[4] and [8], that provide examples of applications of the problem of high-dimensional landscapes optimization in several contexts;
• Modified Fig. 2 and added a reference to the source of the image in the caption;
• Replaced 'matrix denoising' with 'low-rank matrix estimation' throughout the notes, and similarly for the tensor case;
• Modified the discussion on the Jacobian of the change of variable around Eq. (4);
• Modified the discussion on the statistics of eigenvectors of GOE matrices, added a mention to the Haar measure (footnote 2);
• Clarified that the denoising problem is addressed for typical instances of the noise;
• Implemented the changes suggested in points 10-19 of the report;
• Added a discussion on how the joint distribution of eigenvalues and eigenvectors’ projections is obtained, Eq. (57);
• Added comments to clarify the nature of the low-temperature phase of the p=2 spherical p-spin model;
• Clarified that the DMFT analysis is independent of the parameter r, and thus the analysis presented in Ref. [45] and discussed in the notes is valid also for r>0, within the timescales accessible with DMFT;
• Improved the discussion around Eq. (79), on the dependence of the scaling function in the critical regime;
• Corrected the expression for the variance of the symmetric random tensor, Eq. (82). Added footnote 4 to clarify the origin of the combinatorial terms;
• Implemented the changes suggested in points 25 and 26 of the referee's report;
• Improved the justification of Eq. (112);
• Specified that the expression for I(y) given in Eq.(115) holds true for y<0;
• Added a comment on reparametrization invariance regarding Ref. [95];
• Corrected typos identified by the referee (and some more);
• Added references.