Added legend and some small changes.

2 files changed, 39 insertions, 36 deletions

diff --git a/figs/24_detail_letter_legend.pdf b/figs/24_detail_letter_legend.pdf
new file mode 100644
index 0000000..ca32ca0
--- /dev/null
+++ b/figs/24_detail_letter_legend.pdf
diff --git a/frsb_kac-rice_letter.tex b/frsb_kac-rice_letter.tex
index 21807cb..7fc71a9 100644
--- a/frsb_kac-rice_letter.tex
+++ b/frsb_kac-rice_letter.tex
@@ -51,7 +51,7 @@ network.
 
 The \emph{complexity} of a function is the average of the logarithm of the
 number of its minima, maxima, and saddle points (collectively stationary
-points), under conditions like the value of the energy or the index of the
+points), under conditions fixing the value of the energy or the index of the
 stationary point \cite{Bray_1980_Metastable}. Since in complex landscapes this
 number grows exponentially with system size, their complexity is an extensive
 quantity. Understanding the complexity offers an understanding about the
@@ -100,21 +100,20 @@ Cavagna_1997_Structure, Cavagna_1998_Stationary, Cavagna_2005_Cavity,
 Giardina_2005_Supersymmetry}. The landscape complexity has been proven for pure
 and mixed models without RSB \cite{Auffinger_2012_Random,
 Auffinger_2013_Complexity, BenArous_2019_Geometry}. The mixed models been
-treated in specific cases, again without RSB \cite{Folena_2020_Rethinking,
-Ros_2019_Complex}. And the methods of complexity have been used to study many
-geometric properties of the pure models, from the relative position of
-stationary points to one another to shape and prevalence of instantons
-\cite{Ros_2019_Complexity, Ros_2021_Dynamical}.
+treated without RSB \cite{Folena_2020_Rethinking}. And the methods of
+complexity have been used to study many geometric properties of the pure
+models, from the relative position of stationary points to one another to shape
+and prevalence of instantons \cite{Ros_2019_Complexity, Ros_2021_Dynamical}.
 
 The variance of the couplings implies that the covariance of the energy with
 itself depends on only the dot product (or overlap) between two configurations.
 In particular, one finds
 \begin{equation} \label{eq:covariance}
-  \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right)
+  \overline{H(\mathbf s_1)H(\mathbf s_2)}=Nf\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right),
 \end{equation}
 where $f$ is defined by the series
 \begin{equation}
-  f(q)=\frac12\sum_pa_pq^p
+  f(q)=\frac12\sum_pa_pq^p.
 \end{equation}
 One needn't start with a Hamiltonian like
 \eqref{eq:hamiltonian}, defined as a series: instead, the covariance rule
@@ -149,17 +148,17 @@ The stability $\mu$, sometimes called the radial reaction, determines the depth
 of minima or the index of saddles. At large $N$ the Hessian can be shown to
 consist of the sum of a GOE matrix with variance $f''(1)/N$ shifted by a
 constant diagonal matrix of value $\mu$. Therefore, the spectrum of the Hessian
-is a Wigner semicircle of radius $\mu_m=\sqrt{4f''(1)}$ centered at $\mu$. When
-$\mu>\mu_m$, stationary points are minima whose sloppiest eigenvalue is
-$\mu-\mu_m$. When $\mu=\mu_m$, the stationary points are marginal minima with
-flat directions. When $\mu<\mu_m$, the stationary points are saddles with
+is a Wigner semicircle of radius $\mu_\mathrm m=\sqrt{4f''(1)}$ centered at $\mu$. When
+$\mu>\mu_\mathrm m$, stationary points are minima whose sloppiest eigenvalue is
+$\mu-\mu_\mathrm m$. When $\mu=\mu_\mathrm m$, the stationary points are marginal minima with
+flat directions. When $\mu<\mu_\mathrm m$, the stationary points are saddles with
 indexed fixed to within order one (fixed macroscopic index).
 
 It's worth reviewing the complexity for the best-studied case of the pure model
 for $p\geq3$ \cite{Cugliandolo_1993_Analytical}. Here, because the covariance
 is a homogeneous polynomial, $E$ and $\mu$ cannot be fixed separately, and one
 implies the other: $\mu=pE$. Therefore at each energy there is only one kind of
-stationary point. When the energy reaches $E_\mathrm{th}=-\mu_m/p$, the
+stationary point. When the energy reaches $E_\mathrm{th}=-\mu_\mathrm m/p$, the
 population of stationary points suddenly shifts from all saddles to all minima,
 and there is an abrupt percolation transition in the topology of
 constant-energy slices of the landscape. This behavior of the complexity can be
@@ -240,12 +239,14 @@ find the complexity everywhere. This is how the data in what follows was produce
   \centering
   \hspace{-1em}
   \includegraphics[width=\columnwidth]{figs/316_complexity_contour_1_letter.pdf}
+  \includegraphics[width=\columnwidth]{figs/316_detail_letter_legend.pdf}
 
   \caption{
     Complexity of the $3+16$ model in the energy $E$ and stability $\mu$
-    plane. The right shows a detail of the left. Below the yellow marginal line
-    the complexity counts saddles of increasing index as $\mu$ decreases.
-    Above the yellow marginal line the complexity counts minima of increasing
+    plane. Solid lines show the prediction of 1RSB complexity, while dashed
+    lines show the prediction of RS complexity. Below the yellow marginal line
+    the complexity counts saddles of increasing index as $\mu$ decreases. Above
+    the yellow marginal line the complexity counts minima of increasing
     stability as $\mu$ increases.
   } \label{fig:2rsb.contour}
 \end{figure}
@@ -257,7 +258,8 @@ find the complexity everywhere. This is how the data in what follows was produce
 
   \caption{
     Detail of the `phases' of the $3+16$ model complexity as a function of
-    energy and stability. Above the yellow marginal stability line the
+    energy and stability. Solid lines show the prediction of 1RSB complexity, while dashed
+    lines show the prediction of RS complexity. Above the yellow marginal stability line the
     complexity counts saddles of fixed index, while below that line it counts
     minima of fixed stability. The shaded red region shows places where the
     complexity is described by the 1RSB solution, while the shaded gray region
@@ -318,7 +320,7 @@ In this model, the RS complexity gives an inconsistent answer for the
 complexity of the ground state, predicting that the complexity of minima
 vanishes at a higher energy than the complexity of saddles, with both at a
 lower energy than the equilibrium ground state. The 1RSB complexity resolves
-these problems, predicting the same ground state as equilibrium and with a
+these problems, shown in Fig.~\ref{fig:2rsb.contour}. It predicts the same ground state as equilibrium and with a
 ground state stability $\mu_0=6.480\,764\ldots>\mu_\mathrm m$. It predicts that
 the complexity of marginal minima (and therefore all saddles) vanishes at
 $E_\mathrm m$, which is very slightly greater than $E_0$. Saddles become
@@ -356,32 +358,33 @@ model stall in a place where minima are exponentially subdominant.
 \begin{figure}
   \centering
   \includegraphics[width=\columnwidth]{figs/24_phases_letter.pdf}
+  \includegraphics[width=\columnwidth]{figs/24_detail_letter_legend.pdf}
   \caption{
     `Phases' of the complexity for the $2+4$ model in the energy $E$ and
-    stability $\mu$ plane. The region shaded gray shows where the RS solution
-    is correct, while the region shaded red shows that where the FRSB solution
-    is correct. The white region shows where the complexity is zero.
+    stability $\mu$ plane. Solid lines show the prediction of 1RSB complexity,
+    while dashed lines show the prediction of RS complexity. The region shaded
+    gray shows where the RS solution is correct, while the region shaded red
+    shows that where the FRSB solution is correct. The white region shows where
+    the complexity is zero.
   } \label{fig:frsb.phases}
 \end{figure}
 
 If the covariance $f$ is chosen to be concave, then one develops FRSB in equilibrium. To this purpose, we choose
 \begin{equation}
-  f(q)=\frac12\left(q^2+\frac1{16}q^4\right)
-\end{equation}
-also studied before in equilibrium \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical}. Because the ground state is FRSB, for this model
-\begin{equation}
-  E_0=E_\mathrm{alg}=E_\mathrm{th}=-\int_0^1dq\,\sqrt{f''(q)}=-1.059\,384\,319\ldots
+  f(q)=\frac12\left(q^2+\frac1{16}q^4\right),
 \end{equation}
-In the equilibrium solution, the transition temperature from RS to FRSB is $\beta_\infty=1$, with corresponding average energy $\langle E\rangle_\infty=-0.53125\ldots$.
-
-Fig.~\ref{fig:frsb.phases} shows the regions of complexity for the $2+4$ model.
-Notably, the phase boundary predicted by a perturbative expansion
-correctly predicts where the finite $k$RSB approximations terminate.
-Like the 1RSB model in the previous subsection, this phase boundary is oriented
-such that very few, low energy, minima are described by a FRSB solution, while
-relatively high energy saddles of high index are also. Again, this suggests
-that studying the mutual distribution of high-index saddle points might give
-insight into lower-energy symmetry breaking in more general contexts.
+also studied before in equilibrium \cite{Crisanti_2004_Spherical, Crisanti_2006_Spherical}. Because the ground state is FRSB, for this model $E_0=E_\mathrm{alg}=E_\mathrm{th}=E_\mathrm m$.
+In the equilibrium solution, the transition temperature from RS to FRSB is $\beta_\infty=1$, with corresponding average energy $\langle E\rangle_\infty$, also in Table~\ref{tab:energies}.
+
+Fig.~\ref{fig:frsb.phases} shows the regions of complexity for the $2+4$ model,
+computed using finite-$k$ RSB approximations. Notably, the phase boundary
+predicted by a perturbative expansion correctly predicts where the finite
+$k$RSB approximations terminate. Like the 1RSB model in the previous
+subsection, this phase boundary is oriented such that very few, low energy,
+minima are described by a FRSB solution, while relatively high energy saddles
+of high index are also. Again, this suggests that studying the mutual
+distribution of high-index saddle points might give insight into lower-energy
+symmetry breaking in more general contexts.
 
 We have used our solution for mean-field complexity to explore how hierarchical
 RSB in equilibrium corresponds to analogous hierarchical structure in the