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+----------------------------------------------------------------------
+Response to Referee A -- LZ16835/Kent-Dobias
+----------------------------------------------------------------------
+
+Referee A wrote:
+> The authors consider the mean-field p-spin spherical model with
+> *complex* variables and study the number of saddle points of the
+> energy and the eigenvalue distribution of their Hessian matrix. The
+> main result of the rather technical computation is that in a
+> particular limit (concretely kappa->1) the known results for the real
+> p-spin spherical model are reproduced, the (expected) Bézout bound for
+> the number of solutions of the saddle point equations is reached and
+> that the relationships between the “threshold” and extremal state
+> energies is richer in the complex case than in the real case.
+> 
+> I must admit that I was not able to grasp any far-reaching
+> consequences of the computational tour de force only hinted at in the
+> manuscript, and I fear that a nonexpert reader would also not be able
+> to do so. Two arguments are pushed forward by the authors to justify
+> the dissemination of their results to the broader readership of PRL:
+> One is that there are indeed situations in which complex variables
+> appear naturally in disordered system. The first example the authors
+> mention is a Hamiltonian that could be relevant for with random Laser
+> problems and was analyzed 2015 in PRA, which has up to now 30
+> citations according to Google Scholar, and the second example is a
+> Hamiltonian from sting theory that was analyzed in 2016 in JHEP, which
+> has up to now 31 citations. I do not feel that these two examples
+> prove that the enumeration of saddle points if the p-spin model is
+> important or of broad interest.
+> 
+> The second argument of the authors is that extending a real problem to
+> the complex plane often uncovers underlying simplicity that is
+> otherwise hidden, shedding light on the original real problem. Here I
+> come back to what I already mentioned above: I do not see any
+> simplicity emerging from the present calculation and I also do not see
+> the original problem in a new light. Therefore, I do not think that
+> one of the four PRL criteria is actually fulfilled and I recommend to
+> transfer the manuscript to PRE.
+
+We disagree with the referee's assessment here, as we have also explained in
+our letter to the editors. Something in particular that goes unaddressed is
+another motivation (which in the referee's defense we did not enumerate clearly
+in our draft): that understanding the distribution of complex critical points
+is necessary in the treatment of a large class of integrals involved both in
+the definition of quantum mechanics with a complex action and in ameliorating
+the sign problem in, e.g., lattice QCD.
+
+If the criteria for publication is to be "first past the post" of cited
+citations, one might examine our citations of that literature:
+
+  - Analytic continuation of Chern-Simons theory, E Witten (2011): 444 citations
+
+  - New approach to the sign problem in quantum field theories: High density
+    QCD on a Lefschetz thimble, M Cristoforetti et al (2012): 285 citations
+
+Both works are concerned with the location and relative positions of critical
+points of complex theories. In the resubmitted manuscript we have better
+emphasized this motivation.
+
+Referee A wrote:
+> Although the first part of the manuscript is well written and well
+> understandable (at least for me) from page 2 on it becomes very
+> technical and unreadable for a non-expert. If the reader skips to the
+> results and tries to understand the figures she/he is left with the
+> ubiquitous parameter a, whose physical meaning is hidden deep in the
+> saddle point calculation (“dictates the magnitude of |z|^2” – well,
+> with respect to the solutions of (3): is “a” the average value of the
+> modulus squared of the solution z’s or not?). Similar with epsilon:
+> apparently it is the average energy of the saddle point solution – why
+> not writing so also in the figure captions? The paper would profit a
+> lot from a careful rewriting of at least the result section and to
+> provide figure captions with the physical meaning of the quantities
+> and parameters shown.
+
+We thank the referee for their helpful suggestions with regards to the
+readability of our manuscript. In the resubmitted version, much has been
+rewritten for clarity. We would like to highlight several of the most
+substantive changes:
+
+  - The ubiquitous parameter 'a' was replaced by the more descriptive 'r^2', as
+    it is a sort of radius, along with a new parameter 'R^2' which bounds it.
+    Descriptions in English of these were added to the figure captions.
+
+  - The technical portion of the paper was reordered to connect better with the
+    sections preceding and following it.
+
+  - The location of the results is now indicated before the beginning of the
+    technical portion for readers interested in skipping ahead.
+
+Referee A wrote:
+> A couple of minor, technical, quibbles:
+> 
+> 1) If there is any real world application of a p-spin model with
+> complex variables it will NOT have a spherical constraint. I would
+> suggest to discuss the consequences of this constraint, which is
+> introduced for computational simplicity.
+> 
+> 2) After eq. (2): ”We choose to constrain our model by z^2=N.“ Then it
+> is not a spherical constraint any more – does it have any physical
+> relevance?
+
+We have added a more detailed discussion of the constraint to address these
+confusions, emphasizing its purpose. The new paragraphs are:
+
+> One might balk at the constraint $z^Tz=N$---which could appropriately be
+> called a \emph{hyperbolic} constraint---by comparison with $z^\dagger z=N$.
+> The reasoning behind the choice is twofold.
+>
+> First, we seek draw conclusions from our model that are applicable to generic
+> holomorphic functions without any symmetry. Samples of $H_0$ nearly provide
+> this, save for a single anomaly: the value of the energy and its gradient at
+> any point $z$ correlate along the $z$ direction, with $\overline{H_0\partial
+> H_0}\propto \overline{H_0(\partial H_0)^*}\propto z$. This anomalous
+> direction should thus be forbidden, and the constraint surface $z^Tz=N$
+> accomplishes this.
+>
+> Second, taking the constraint to be the level set of a holomorphic function
+> means the resulting configuration space is a \emph{bone fide} complex
+> manifold, and therefore permits easy generalization of the integration
+> techniques referenced above. The same cannot be said for the space defined by
+> $z^\dagger z=N$, which is topologically the $(2N-1)$-sphere and cannot admit
+> a complex structure.
+>
+> Imposing the constraint with  a holomorphic function makes the resulting
+> configuration space a \emph{bone fide} complex manifold, which is, as we
+> mentioned, the situation we wish to model. The same cannot be said for the
+> space defined by $z^\dagger z=N$, which is topologically the $(2N-1)$-sphere,
+> does not admit a complex structure, and thus yields a trivial structure of
+> saddles.  However, we will introduce the bound $r^2\equiv z^\dagger z/N\leq
+> R^2$ on the `radius' per spin as a device to classify saddles.   We shall see
+> that this `radius' $r$ and its upper bound $R$ are insightful knobs in our
+> present problem, revealing structure as they are varied. Note that taking
+> $R=1$ reduces the problem to that of the ordinary $p$-spin.
+
+Referee A wrote:
+> 3) On p.2: “…a, which dictates the magnitude of |z|^2, or
+> alternatively the magnitude y^2 of the imaginary part. The last part
+> is hard to understand, should be explained.
+
+We thank the referee for pointing out this confusing statement, which was
+unnecessary and removed.
+
+> 4) On p.2: “In most the parameter space we shall study her, the
+> annealed approximation is exact.” I think it is necessary to provide
+> some evidence her, because the annealed approximation is usually a
+> pretty severe approximation.
+
+We have nuanced the statement in question and added a citation to a review
+article which outlines the reasoning for analogous models. The amended sentence
+reads:
+
+> Based on the experience from similar problems \cite{Castellani_2005_Spin-glass},
+> the \emph{annealed approximation} $N\Sigma\sim\log\overline{\mathcal N}$ is
+> expected to be exact wherever the complexity is positive.
+
+Sincerely,
+Jaron Kent-Dobias & Jorge Kurchan
+