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diff --git a/referee_respose.txt b/referee_respose.txt new file mode 100644 index 0000000..3b204c6 --- /dev/null +++ b/referee_respose.txt @@ -0,0 +1,158 @@ +---------------------------------------------------------------------- +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 + |