Update on Overleaf.

2 files changed, 67 insertions, 49 deletions

diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib
index 20d5a32..b1922a0 100644
--- a/frsb_kac-rice.bib
+++ b/frsb_kac-rice.bib
@@ -197,6 +197,16 @@
   year={1992},
   publisher={APS}
 }
+@article{mezard1992manifolds,
+  title={Manifolds in random media: two extreme cases},
+  author={M{\'e}zard, Marc and Parisi, Giorgio},
+  journal={Journal de Physique I},
+  volume={2},
+  number={12},
+  pages={2231--2242},
+  year={1992},
+  publisher={EDP Sciences}
+}
 
 @article{crisanti1995thouless,
   title={Thouless-Anderson-Palmer approach to the spherical p-spin spin glass model},
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 1366988..8086c70 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -23,59 +23,18 @@ in fact  the problem has been open
 ever since.
 Indeed, to this date the program of computing the number of saddles of a mean-field 
 glass has been only  carried out for a small subset of models.
-These include most notable the p-spin model ($p>2$) \cite{rieger1992number,crisanti1995thouless}
-In this paper we present what we believe is the general ansatz for the solution.
-It includes the Parisi solution as the limit of lowest states, as it should.
+These include most notably the p-spin model ($p>2$) \cite{rieger1992number,crisanti1995thouless}. 
+The problem of studying the critical points of these landscapes
+has evolved into an active field in probability theory \cite{Auffinger_2012_Random,Auffinger_2013_Complexity,BenArous_2019_Geometry}
 
+In this paper we present what we believe is the general ansatz for the 
+computation of saddles of generic mean-field models,  including the Sherrington-Kirkpatrick model. It incorporates  the Parisi solution as the limit of lowest states, as it should.
 
-\subsection{What to expect?}
-
-In order to try to visualize what one should expect, consider two pure p-spin models, with 
-\begin{equation}
-    H = H_1 + H_2=\alpha_1 \sum_{ijk} J^1_{ijk} s_i s_j s_k +
-    \alpha_2 \sum_{ijk} J^2_{ijk} \bar s_i \bar s_j \bar s_k +\epsilon \sum_i s_i \bar s_i
-\end{equation}
-The complexity of the first and second systems in terms of $H_1$ and of $H_2$ 
-have, in the absence of coupling,  the same dependence, but are stretched to one another:
-\begin{equation}
-    \Sigma_1(H_1)= \Sigma_o(H_1/\alpha_1) \qquad ; \qquad \Sigma_2(H_2)= \Sigma_o(H_2/\alpha_2) 
-\end{equation}
-Each system has a ground state energy $E_{gs}^{1,2}$, a threshold energy $E_{thres}^{1,2}$ (a well-defined notion, since we are considering pure p-spins), abd the corresponding limit values $X^{1,2}_{gs}=\left. \frac{d \Sigma_1}{dE_{1,2}}\right|_{E^{gs}_{12}}$
-and $X^{1,2}_{thres}=\left. \frac{d \Sigma_1}{dE_{1,2}}\right|_{E^{thres}_{12}}$
-Considering the cartesian product of both systems, we have, in terms of the total energy
-$H=H_1+H_2$ three regimes:
-\begin{itemize}
-\item  {\bf Unfrozen}: 
-\begin{eqnarray}
-& & X_1 \equiv \frac{d \Sigma_1}{dE_1}=  X_2 \equiv \frac{d \Sigma_2}{dE_2}
- \end{eqnarray}
-\item  {\bf Semi-frozen}
-As we go down in energy, one of the systems (say, the first) reaches its ground state,
-At lower temperatures, the first system is thus frozen, while the second is not,
-so that $X_1=X_1^{gs}> X_2$. The lowest energy is such that both systems are frozen.
-\item {\bf Semi-threshold }  As we go up from the unfrozen upwards in energy,
-the second system reaches its threshold $X_2^{thres}$. At higher energies minima are extremely rare,
-so the second system remains stuck at its threshold for higher energies.
-
-\item{\bf Both systems reach their thresholds} There essentially no more minima above that.
-\end{itemize}
-Consider now two combined vectors $({\bf s},{\bf \hat s})$ and $({\bf s}',{\bf \hat s}')$
-chosen at the same energies.\\
-
-$\bullet$ Their normalized overlap is close to one when both subsystems are frozen,
-close to a half in the semifrozen phase, and zero at all higher energies.\\
-
-$\bullet$ In phases where one or both systems are stuck in their thresholds (and only in those), the
-minima are exponentially subdominant with respect to saddles.
-
-$\bullet$ {\bf note that the same reasoning leads us to the conclusion that
-minima of two total energies such that one of the systems is frozen have nonzero overlaps}
 
 \section{The model}
 
-Here we consider, for definiteness, the `toy' model introduced by M\'ezard and Parisi (????? what)
-
-The mixed $p$-spin model
+Here we consider, for definiteness, the  mixed $p$-spin model, itself a particular case 
+of the `Toy Model' of M\'ezard and Parisi \cite{mezard1992manifolds}
 \begin{equation}
   H(s)=\sum_p\frac{a_p^{1/2}}{p!}\sum_{i_1\cdots i_p}J_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
 \end{equation}
@@ -116,6 +75,49 @@ points are minima whose sloppiest eigenvalue is $\mu-\mu_m$. Finally,
 when $\mu=\mu_m$, the critical points are marginal minima.
 
 
+\subsection{What to expect?}
+
+In order to try to visualize what one should expect, consider two pure p-spin models, with 
+\begin{equation}
+    H = H_1 + H_2=\alpha_1 \sum_{ijk} J^1_{ijk} s_i s_j s_k +
+    \alpha_2 \sum_{ijk} J^2_{ijk} \bar s_i \bar s_j \bar s_k +\epsilon \sum_i s_i \bar s_i
+\end{equation}
+The complexity of the first and second systems in terms of $H_1$ and of $H_2$ 
+have, in the absence of coupling,  the same dependence, but are stretched to one another:
+\begin{equation}
+    \Sigma_1(H_1)= \Sigma_o(H_1/\alpha_1) \qquad ; \qquad \Sigma_2(H_2)= \Sigma_o(H_2/\alpha_2) 
+\end{equation}
+Each system has a ground state energy $E_{gs}^{1,2}$, a threshold energy $E_{thres}^{1,2}$ (a well-defined notion, since we are considering pure p-spins), abd the corresponding limit values $X^{1,2}_{gs}=\left. \frac{d \Sigma_1}{dE_{1,2}}\right|_{E^{gs}_{12}}$
+and $X^{1,2}_{thres}=\left. \frac{d \Sigma_1}{dE_{1,2}}\right|_{E^{thres}_{12}}$
+Considering the cartesian product of both systems, we have, in terms of the total energy
+$H=H_1+H_2$ three regimes:
+\begin{itemize}
+\item  {\bf Unfrozen}: 
+\begin{eqnarray}
+& & X_1 \equiv \frac{d \Sigma_1}{dE_1}=  X_2 \equiv \frac{d \Sigma_2}{dE_2}
+ \end{eqnarray}
+\item  {\bf Semi-frozen}
+As we go down in energy, one of the systems (say, the first) reaches its ground state,
+At lower temperatures, the first system is thus frozen, while the second is not,
+so that $X_1=X_1^{gs}> X_2$. The lowest energy is such that both systems are frozen.
+\item {\bf Semi-threshold }  As we go up from the unfrozen upwards in energy,
+the second system reaches its threshold $X_2^{thres}$. At higher energies minima are extremely rare,
+so the second system remains stuck at its threshold for higher energies.
+
+\item{\bf Both systems reach their thresholds} There essentially no more minima above that.
+\end{itemize}
+Consider now two combined vectors $({\bf s},{\bf \hat s})$ and $({\bf s}',{\bf \hat s}')$
+chosen at the same energies.\\
+
+$\bullet$ Their normalized overlap is close to one when both subsystems are frozen,
+close to a half in the semifrozen phase, and zero at all higher energies.\\
+
+$\bullet$ In phases where one or both systems are stuck in their thresholds (and only in those), the
+minima are exponentially subdominant with respect to saddles.
+
+$\bullet$ {\bf note that the same reasoning leads us to the conclusion that
+minima of two total energies such that one of the systems is frozen have nonzero overlaps}
+
 \section{Main result}
 
 \begin{equation}
@@ -294,6 +296,11 @@ $F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $y$ and shifted by $z$.
   \Sigma(\epsilon,\mu)=\frac1N\log\mathcal N(\epsilon, \mu)
 \end{equation}
 
+{\em The `mass' term $\mu$ may take a fixed value, or it may be an integration constant,
+for example fixing the spherical constraint.
+This will turn out to be important when we discriminate between counting all solutions, or selecting those of a given index, for example minima}
+
+
 \subsection{The replicated  problem}
 
 \cite{Ros_2019_Complex}
@@ -316,7 +323,8 @@ the question of independence \cite{Bray_2007_Statistics}
     \times
     \overline{\prod_a^n |\det(\partial\partial H(s_a)-\mu I)|}
   \end{aligned}
-\end{equation}
+\end{equation}{\bf The average breaks into a product of two independent averages, one for the gradient
+and one for the determinant. The integration of all variables may be restricted to the domain such that the matrix $\partial\partial H(s_a)-\mu I$ has  a specified number of negative eigenvalues Fyodorov? and }
 
 \begin{equation}
   \begin{aligned}