\documentclass[a4paper]{letter} \usepackage[utf8]{inputenc} % why not type "Bézout" with unicode? \usepackage[T1]{fontenc} % vector fonts plz \usepackage{newtxtext,newtxmath} % Times for PR \usepackage[ colorlinks=true, urlcolor=purple, linkcolor=black, citecolor=black, filecolor=black, ]{hyperref} % ref and cite links with pretty colors \usepackage{xcolor} \usepackage[style=phys]{biblatex} \renewcommand{\thefootnote}{\fnsymbol{footnote}} \addbibresource{frsb_kac-rice.bib} \signature{ \vspace{-6\medskipamount} \smallskip Jaron Kent-Dobias \& Jorge Kurchan } \address{ Laboratoire de Physique\\ Ecole Normale Sup\'erieure\\ 24 rue Lhomond\\ 75005 Paris } \begin{document} \begin{letter}{ Agnese I.~Curatolo, Ph.D.\\ Physical Review Letters\\ 1 Research Road\\ Ridge, NY 11961 } \opening{Dear Dr.~Curatolo,} Enclosed please find a revised manuscript. Neither referee criticized the scientific content of our paper, nor substantively addressed its presentation. We have followed their comments in the direction of highlighting the importance of having a full solution. In particular we have emphasized that going to the full replica treatment uncovers a phase-space structure that needs to be taken into account, and that is absent in the annealed treatment. Among other changes, we have added the paragraph: \begin{quote} Having a full, exact (`quenched') solution of the generic problem is not primarily a matter of {\em accuracy}. Very basic structural questions are omitted in the approximate `annealed' solution. What is lost is the nature, at any given energy (or free energy) level, of the stationary points in a generic energy function: at low energies are they basically all minima, with an exponentially small number of saddles, or -- as we show here -- do they consist of a mixture of saddles whose index -- the number of unstable directions -- is a smoothly distributed number? These questions need to be answered for the understanding of the relevance of more complex objects such as barrier crossing (which barriers?) \footfullcite{Ros_2019_Complexity, Ros_2021_Dynamical}, or the fate of long-time dynamics (which end in what kind of target states?). \end{quote} Both referees find that our paper is clearly written but technical, and that its topic of ``the different RSB schemes'' are not suitable for a broad audience. This is surprising to the authors, since a quick search on Google Scholar reveals several recent PRLs with heavy use of RSB schemes. It is perhaps true that the final solution of an open problem may often be more technical than the previous ones. But we would like to submit to the referees that it is somewhat incongruous that the solution to a problem that had remained open for 42 years -- during which it was always present in articles in PRL \footfullcite{Fyodorov_2004_Complexity, Bray_2007_Statistics, Fyodorov_2012_Critical, Wainrib_2013_Topological, Dennis_2020_Jamming}-- is rejected because it demands of the readers a slightly longer attention span. These previous works were often limited by the fact that general landscapes (for which an annealed solution is not exact) were inaccessible. Below, we respond to the referees' comments. A comprehensive accounting of the changes to our manuscript can be found appended to this letter. \begin{quote} \begin{center} Report of Referee A -- LY17256/Kent-Dobias \end{center} \it The authors consider spin glass models with mixed p-spin interactions on the N-Sphere and calculate the number of stationary points, the logarithm of which yields the complexity. The disorder average of this logarithm is computed with the replica trick, and for different model variants different replica symmetry breaking (RSB) solutions are obtained. A new feature of the solutions, in contrast to previous replica symmetric calculations, is that RSB must occur in parts of the energy-stability phase diagram. \hspace{2em}The paper is clearly written although the content is rather technical and probably only accessible to experts in mean field spin glass models and the different RSB schemes developed in this field. In connection with the well-studied p=3 spin glass model it is briefly mentioned that the complexity and its transitions as a function of energy and/or stability is relevant for the equilibrium and the dynamical behavior of this model – but such a connection has not been made here. \hspace{2em}Therefore, I feel that the results presented here are only interesting for group of experts and I do not assess the finding that the complexity of mixed p-spin glass models shows RSB as a major breakthrough in the field. Therefore, the manuscript is not suitable for publication in Phys.\ Rev.\ Lett., and the publication of the accompanying longer paper, submitted to PRE, is sufficient to disseminate the results summarized in this manuscript. \end{quote} Referee A correctly points out that one new feature of the solutions outlined in our manuscript is that RSB must occur in parts of the phase diagram for these models they are indeed the solutions of the \textit{quenched} complexity, which has not been correctly calculated until now. We agree with the referee that ``the complexity of the mixed p-spin glass models'' is not a major breakthrough in and of itself, we just chose to demonstrate the problem in simplest toy model. But believe that the technique for computing the quenched complexity is a major breakthrough \textit{because it brings in the features of organization of saddles of all kinds that are invisible in the annealed scheme}. Referee A states that a connection between the complexity and the equilibrium and dynamical behavior is not made in our paper. Until recently, this connection was taken for granted, and the demonstration that the standard correspondence does not hold in the mixed p-spin spherical models was exciting enough news to be published in PRX 10, 031045 (2020). It is true that our work doesn't solve the problem that paper opened, but it does deepen it by showing definitively that the use of RSB and the quenched complexity are not sufficient to reestablish a landscape–dynamics connection. {\bf One can hardly expect that the structure of saddles at a given energy may be connected with dynamics (for example in Sherrington Kirkpatrick) if it is unknown}. \begin{quote} \begin{center} Report of Referee B -- LY17256/Kent-Dobias \end{center} \textit{The paper presents a computation of the complexity in spherical spin-glass models. Neither the techniques nor the results are sufficiently new and relevant to justify publication on PRL. This is not surprising given that the topic has been studied extensively in the last thirty years and more, the only novelty with respect to previous work is that the results are obtained at zero temperature but this is definitively not enough. Essential open problems in the field involves dynamics and activated processes and some results have appeared recently, instead the analysis of the static landscape, to which the present paper is a variation, failed to deliver answers to these questions up to now. } \end{quote} Concerning the statement of Referee B that ``the only novelty with respect to previous work is that the results are obtained at zero temperature,'' we do not know what to make of the referee's statement. The novelty of the paper is most definitely not the fact of treating a zero temperature case. We have added the following phrase, that should clarify the situation: \begin{quote} For simplicity we have concentrated here on the energy, rather than {\em free-energy} landscape, although the latter is sometimes more appropriate. From the technical point of view, this makes no fundamental difference, it suffices to apply the same computation to the Thouless-Andreson-Palmer (TAP) free energy, \footfullcite{Crisanti_1995_Thouless-Anderson-Palmer} instead of the energy. We do not expect new features or technical complications arise. \end{quote} We agree with Referee B's assessment of ``essential open problems in the field,'' and agree that our work does not deliver all answers. However, delivering all answers for all essential open problems is not the acceptance criterion of PRL. Our manuscript makes essential steps toward solving the critical problem of connecting analysis of the static landscape to dynamics. We believe that its essential step is through the introduction of a new technique, calculation of the quenched complexity, which we believe will have significant impact as it is applied to more complicated models. \closing{Sincerely,} \vspace{1em} \end{letter} \end{document}