Lots of writing tweaks.

1 files changed, 169 insertions, 122 deletions

diff --git a/2-point.tex b/2-point.tex
index 21d7dbf..443602a 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -110,15 +110,19 @@
 
 Many systems exhibit ``glassiness,'' characterized by rapid slowing of dynamics
 over a short parameter interval. These include actual (structural) glasses,
-spin glasses, certain inference and optimization problems, and more. Glassiness is qualitatively understood to arise from structure of
-an energy or cost landscape, whether due to the proliferation of metastable
-states, or to the raising of barriers which cause effective dynamic constraints \cite{Cavagna_2001_Fragile, Stillinger_2013_Glass, Kirkpatrick_2015_Colloquium}.
-However, in most models there is no known quantitative correspondence between these
-landscape properties and the dynamic behavior they are purported to describe.
+spin glasses, certain inference and optimization problems, and more. Glassiness
+is qualitatively understood to arise from structure of an energy or cost
+landscape, whether due to the proliferation of metastable states, or to the
+raising of barriers which cause effective dynamic constraints
+\cite{Cavagna_2001_Fragile, Stillinger_2013_Glass,
+Kirkpatrick_2015_Colloquium}. However, in most models there is no known
+quantitative correspondence between these landscape properties and the dynamic
+behavior they are purported to describe.
 
 There is such a correspondence in one of the simplest mean-field model of
 glasses: in the pure spherical models, the dynamic transition corresponds with
-the energy level at which thermodynamic states attached to marginal inherent states\footnote{
+the energy level at which thermodynamic states attached to marginal inherent
+states\footnote{
   For this paper, which focuses on minima, we will take \emph{state} to mean
   \emph{minimum} or equivalently \emph{inherent state} and not a thermodynamic
   state. Any discussion of thermodynamic or equilibrium states will explicitly
@@ -136,11 +140,12 @@ In slightly less simple models, the mixed spherical models, the story changes.
 There are now a range of energies with exponentially many marginal minima. It
 was believed that the energy level at which these marginal minima are the most
 common type of stationary point would play the same role as the threshold
-energy in the pure models. However, recent work has shown that this is
-incorrect. Quenches from different starting temperatures above the dynamical
-transition temperature result in dynamics that approach different energy
-levels, and the purported threshold does not attract the long-time dynamics in
-most cases \cite{Folena_2020_Rethinking, Folena_2021_Gradient}.
+energy in the pure models (in fact we will refer to this energy level as the
+threshold energy in the mixed models). However, recent work has shown that
+this is incorrect. Quenches from different starting temperatures above the
+dynamical transition temperature result in dynamics that approach different
+energy levels, and the purported threshold does not attract the long-time
+dynamics in most cases \cite{Folena_2020_Rethinking, Folena_2021_Gradient}.
 
 This paper studies the two-point structure of stationary points in the mixed
 spherical models, or their arrangement relative to each other, previously
@@ -156,7 +161,7 @@ Then, we compute the logarithm of the number of other points constrained to lie
 at a fixed overlap from the reference point. The fact of constraining the count
 to a fixed overlap produces constrained points with atypical properties. For
 instance, we will see that when the constrained overlap is made sufficiently
-large, the points so constrained tend to have an isolated eigenvalue pulled out
+large, the constrained points tend to have an isolated eigenvalue pulled out
 of their spectrum, and the associated eigenvector is correlated with the
 direction of the reference point. Without the proximity constraint, such an
 isolated eigenvalue amounts to a large deviation from the typical spectrum of
@@ -168,17 +173,19 @@ anything interesting to differentiate sets of them from each other. Though we
 find rich structure in this population, their properties pivot around the
 debunked threshold energy, and the apparent attractors of long-time dynamics
 are not distinguished by this measure. Moreover, we show that the usual picture of a
-marginal `manifold' of inherent states separated by subextensive barriers is only true
+marginal `manifold' of inherent states separated by subextensive barriers \cite{Kurchan_1996_Phase} is only true
 at the threshold energy, while at other energies marginal minima are far apart
-and separated by extensive barriers \cite{Kurchan_1996_Phase}. Therefore, with respect to the problem of
+and separated by extensive barriers. Therefore, with respect to the problem of
 dynamics this paper merely deepens the outstanding issues.
 
 In \S\ref{sec:model}, we introduce the mixed spherical models and discuss their
-properties. In \S\ref{sec:results}, we share the main results of the paper. In
+properties, defining many of the symbols and concepts relied on in the rest of
+the paper. In \S\ref{sec:results}, we share the main results of the paper, with
+a large discussion of the neighborhood of marginal states. In
 \S\ref{sec:complexity} we detail the calculation of the two-point complexity,
 and in \S\ref{sec:eigenvalue} and \S\ref{sec:franz-parisi} we do the same for
 the properties of an isolated eigenvalue and for the zero-temperature
-Franz--Parisi potential.
+Franz--Parisi potential. Finally in \S\ref{sec:conclusion} we draw some conclusions from the study.
 
 \section{Model}
 \label{sec:model}
@@ -409,12 +416,12 @@ than in the pure models, with different types of marginal states being found at
 different energies. Second, these states attract the dynamics (as evidenced by power-law relaxations), and so are the
 inevitable end-point of equilibrium and algorithmic processes \cite{Folena_2023_On}. We find,
 surprisingly, that the properties of marginal states pivot around the threshold
-energy.
+energy, the energy at which most stationary points are marginal.
 
 \begin{itemize}
   \item \textbf{Energies below the threshold.} Marginal states have a
     macroscopic gap in their overlap with nearby minima and saddles. The
-    nearest stationary points are saddles with a single downward direction,
+    nearest stationary points are saddles with an oriented direction,
     and always have a higher energy density than the reference state.
     Fig.~\ref{fig:marginal.prop.below} shows examples of the neighborhoods of
     these marginal minima.
@@ -432,7 +439,7 @@ energy.
     stationary points at arbitrarily close distance, with a cubic pseudogap
     in their complexity. The nearest ones include oriented saddle
     points with an extensive number of downward directions, and oriented stable
-    an marginal minima. Though most of the nearest states are found at higher
+    and marginal minima. Though most of the nearest states are found at higher
     energies, they can be found at the same energy density as the reference
     state. Fig.~\ref{fig:marginal.prop.thres} shows examples of the
     neighborhoods of these marginal states.
@@ -527,8 +534,15 @@ This has implications for how quench dynamics should be interpreted. When
 marginal states are approached above the threshold energy, they must have been
 via the neighborhood of saddles with an extensive index, not other marginal
 states. On the other hand, marginal states approached below the threshold
-energy must  be reached after an extensive distance in
-configuration space without encountering any stationary point. A version of
+energy must  be reached after an extensive distance in configuration space
+without encountering any stationary point. The geometric conditions of the
+neighborhoods above and below are quite different, but the observed aging
+dynamics don't appear to qualitatively change \cite{Folena_2020_Rethinking,
+Folena_2021_Gradient}. Therefore, the conditions in the neighborhood of the
+marginal minimum eventually reached at infinite time appear to be irrelevant
+for the nature of aging dynamics at any finite time.
+
+A version of
 this story was told a long time ago by the authors of
 \cite{Kurchan_1996_Phase}, who write on aging in the pure spherical models
 where the limit of $N\to\infty$ is taken before that of $t\to\infty$: ``it is
@@ -673,7 +687,7 @@ Here the $\hat\beta$s are the fields auxiliary to the energy constraints, the
 $\hat\mu$s are auxiliary to the spherical and overlap constraints, and the
 $\hat{\pmb\sigma}$s and $\hat{\mathbf s}$s are auxiliary to the constraint that
 the gradient be zero.
-We have written the $H$-dependant terms in this strange form for the ease of taking the average over $H$: since it is Gaussian-correlated, it follows that
+We have written the $H$-dependent terms in this strange form for the ease of taking the average over $H$: since it is Gaussian-correlated, it follows that
 \begin{equation}
   \overline{e^{\int d\mathbf t\,\mathcal O(\mathbf t)H(\mathbf t)}}
   =e^{\frac12\int d\mathbf t\,d\mathbf t'\,\mathcal O(\mathbf t)\mathcal O(\mathbf t')\overline{H(\mathbf t)H(\mathbf t')}}
@@ -703,7 +717,7 @@ definitions, integrated over the new matrix fields, which is equivalent to
 multiplying by one. Once this is done, the many scalar products appearing
 throughout can be replaced by the matrix fields, and the original vector fields
 can be integrated over. Conjugate matrix field integrals created when the
-$\delta$ functions are promoted to exponentials can be evaluated by saddle
+$\delta$-functions are promoted to exponentials can be evaluated by saddle
 point in the standard way, yielding an effective action depending on the above
 matrix fields alone.
 
@@ -793,7 +807,7 @@ be a sensible covariance, both $u_f$ and $v_f$ are strictly positive. Note also
 that $u_f=v_f=0$ if $f$ is a homogeneous polynomial as in the pure models.
 These expressions are invalid for the pure models because $\mu_0$ and $E_0$
 cannot be fixed independently; we would have done the equivalent of inserting
-two identical $\delta$-functions! Instead, the terms $\hat\beta_0$ and
+two identical $\delta$-functions. For the pure models, the terms $\hat\beta_0$ and
 $\hat\beta_1$ must be set to zero in our prior formulae (as if the energy was
 not constrained) and then the saddle point taken.
 
@@ -843,8 +857,9 @@ complexities, they have the following form at the saddle point:
     0&\cdots&0
   \end{bmatrix}
 \end{align}
-where only the first row is nonzero as a result of the sole linear term
-proportional to $C_{1b}^{01}$ in the action.
+where only the first row is nonzero. The other entries, which correspond to the
+completely uncorrelated replicas in an \textsc{rsb} picture, are all zero
+because uncorrelated vectors on the sphere are orthogonal.
 
 The inverse of block hierarchical matrix is still a block hierarchical matrix, since
 \begin{equation}
@@ -857,13 +872,14 @@ The inverse of block hierarchical matrix is still a block hierarchical matrix, s
     -i(C^{00}D^{00}+R^{00}R^{00})^{-1}R^{00} & (C^{00}D^{00}+R^{00}R^{00})^{-1}C^{00}
   \end{bmatrix}
 \end{equation}
-Because of the structure of the 01 matrices, the volume element will depend only on the diagonal if this matrix. If we write
+Because of the structure of the 01 matrices, the volume element will depend
+only on the diagonals of the matrices in this inverse block matrix. If we define
 \begin{align}
   \tilde c_\mathrm d^{00}&=[(C^{00}D^{00}+R^{00}R^{00})^{-1}C^{00}]_{\text d} \\
   \tilde r_\mathrm d^{00}&=[(C^{00}D^{00}+R^{00}R^{00})^{-1}R^{00}]_{\text d} \\
   \tilde d_\mathrm d^{00}&=[(C^{00}D^{00}+R^{00}R^{00})^{-1}D^{00}]_{\text d}
 \end{align}
-then the result is
+as the diagonals of the blocks of the inverse matrix, then the result  of the product is
 \begin{equation}
   \begin{aligned}
      &   \begin{bmatrix}
@@ -886,10 +902,9 @@ then the result is
      \end{bmatrix}
   \end{aligned}
 \end{equation}
-where each block is a constant $n\times n$ matrix.
-
-
-Because the matrices $C^{00}$, $R^{00}$, and $D^{00}$ are diagonal in this case, the diagonals of the inverse block matrix from above are simple expressions:
+where each block is a constant $n\times n$ matrix. Because the matrices
+$C^{00}$, $R^{00}$, and $D^{00}$ are diagonal in the replica symmetric case,
+the diagonals of the blocks above take a simple form:
 \begin{align}
   \tilde c_\mathrm d^{00}=f'(1) &&
   \tilde r_\mathrm d^{00}=r^{00}_\mathrm df'(1) &&
@@ -924,41 +939,68 @@ $r^{11}_\mathrm d$, $r^{11}_0$, and $q^{11}_0$, is
   \Bigg]\Bigg\}
   \end{aligned}
 \end{equation}
+It is possible to further extremize this expression over all the other
+variables but $q_0^{11}$, for which the saddle point conditions have a unique
+solution. However, the resulting expression is quite complicated and provides
+no insight. In practice, the complexity can be calculated in two ways. First,
+the extremal problem can be done numerically, initializing from $q=0$ where the
+problem reduces to that of the single-point complexity of points with energy
+$E_1$ and stability $\mu_1$, and then taking small steps in $q$ or other
+parameters to trace out the solution. This is how the data in all the plots of
+this paper was produced. Second, the complexity can be calculated in the near
+neighborhood of a reference point by expanding in small $1-q$. This is what we
+describe in the next subsection.
 
 \subsection{Expansion in the near neighborhood}
 
-The most common neighbors of a reference point are given by further extremizing
-the two-point complexity over the energy $E_1$ and stability $\mu_1$ of the
-nearby points. This gives the conditions
+If there is no overlap gap between the reference point and its nearest
+neighbors, their complexity can be calculated by an expansion in $1-q$. First,
+we'll use this method to describe the most common type of stationary point in
+the close vicinity of a reference point. The most common neighbors of a
+reference point are given by further maximizing the two-point complexity over
+the energy $E_1$ and stability $\mu_1$ of the nearby points. This gives the
+conditions
 \begin{align}
   \hat\beta_1=0 &&
   \mu_1=2r^{11}_\mathrm df''(1)
 \end{align}
 where the second is only true for $\mu_1^2\leq\mu_\mathrm m^2$, i.e., when the
-nearby points are saddle points. Under the conditions where stationary points
-can be found arbitrarily close to their neighbors, we can produce explicit
-formulae for the complexity and the properties of the most common neighbors by
-expanding in powers of $\Delta q=1-q$. For the complexity, the result is
+nearby points are saddle points or marginal minima. When these conditions are
+inserted into the complexity, an expansion is made in small $1-q$, and the
+saddle point in the remaining parameters is taken, the result is
 \begin{equation}
-  \Sigma_{12}=\frac{f'''(1)}{8f''(1)^2}(\mu_\mathrm m^2-\mu_0^2)\left(\sqrt{2+\frac{2f''(1)(f''(1)-f'(1))}{f'''(1)f'(1)}}-1\right)(1-q)
+  \Sigma_{12}
+  =\frac{f'''(1)}{8f''(1)^2}(\mu_\mathrm m^2-\mu_0^2)\left(\sqrt{2+\frac{2f''(1)\big(f''(1)-f'(1)\big)}{f'''(1)f'(1)}}-1\right)(1-q)
   +O\big((1-q)^2\big)
 \end{equation}
-The population of stationary points that are most common at each energy have the relation
+independent of $E_0$. To describe the properties of these most common
+neighbors, it is convenient to first make a definition. The population of
+stationary points that are most common at each energy (the blue line in
+Fig.~\ref{fig:complexities}) have the relation
 \begin{equation}
   E_\mathrm{dom}(\mu_0)=-\frac{f'(1)^2+f(1)\big(f''(1)-f'(1)\big)}{2f''(1)f'(1)}\mu_0
 \end{equation}
-between $E_0$ and $\mu_0$ for $\mu_0^2\leq\mu_\mathrm m^2$. Using this most common value, the energy and stability of the most common neighbors at small $\Delta q$ are
+between $E_0$ and $\mu_0$ for $\mu_0^2\leq\mu_\mathrm m^2$. Using this
+definition, the energy and stability of the most common neighbors at small
+$\Delta q$ are
 \begin{align} \label{eq:expansion.E.1}
   E_1&=E_0+\frac12\frac{v_f}{u_f}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)^2+O\big((1-q)^3\big) \\
     \label{eq:expansion.mu.1}
   \mu_1&=\mu_0-\frac{v_f}{u_f}\big(E_0-E_\mathrm{dom}(\mu_0)\big)(1-q)+O\big((1-q)^2\big)
 \end{align}
-Therefore, whether the energy and stability of nearby points increases or
-decreases from that of the reference point depends only on whether the energy
-of the reference point is above or below that of the most common population at
-the same stability. In particular, since $E_\mathrm{dom}(\mu_\mathrm
-m)=E_\mathrm{th}$, the threshold energy is also the pivot around which the
-points asymptotically nearby marginal minima change their properties.
+The most common neighboring saddles to a reference saddle are much nearer to
+the reference in energy ($\Delta q^2$) than in stability ($\Delta q$). In fact,
+this scaling also holds for the entire range of neighbors to a reference
+saddle, with the limits in energy scaling like $\Delta q^2$ and those of
+stability scaling like $\Delta q$.
+
+Because both expressions are proportional to $E_0-E_\mathrm{dom}(\mu_0)$,
+whether the energy and stability of nearby points increases or decreases from
+that of the reference point depends only on whether the energy of the reference
+point is above or below that of the most common population at the same
+stability. In particular, since $E_\mathrm{dom}(\mu_\mathrm m)=E_\mathrm{th}$,
+the threshold energy is also the pivot around which the points asymptotically
+nearby marginal minima change their properties.
 
 To examine better the population of marginal points, it is necessary to look at
 the next term in the series of the complexity with $\Delta q$, since the linear
@@ -977,12 +1019,30 @@ Note that this expression is only true for $\mu=\mu_\mathrm m$. Therefore,
 among marginal minima, when $E_0$ is greater than the threshold one finds
 neighbors at arbitrarily close distance. When $E_0$ is less than the threshold,
 the complexity of nearby points is negative, and there is a desert where none
-are found. The properties of the nearby states above the threshold can be
+are found.
+
+\begin{figure}
+  \centering
+  \includegraphics{figs/expansion_energy.pdf}
+  \hspace{1em}
+  \includegraphics{figs/expansion_stability.pdf}
+
+  \caption{
+    Demonstration of the convergence of the $(1-q)$-expansion for marginal
+    reference minima. Solid lines and shaded region show are the same as in
+    Fig.~\ref{fig:marginal.prop.above} for $E_0-E_\mathrm{th}\simeq0.00667$.
+    The dotted lines show the expansion of most common neighbors, while the
+    dashed lines in both plots show the expansion for the minimum and maximum
+    energies and stabilities found at given $q$.
+  } \label{fig:expansion}
+\end{figure}
+
+The properties of the nearby states above the threshold can be
 further quantified. The most common points are still given by
 \eqref{eq:expansion.E.1} and \eqref{eq:expansion.mu.1}, but the range of
 available points can also be computed, and one finds that the stability lies in
 the range
-$\mu_1=\mu_0+\delta\mu_1(1-q)\pm\delta\mu_2(1-q)^{3/2}+O\big((1-q)^2\big)$
+$\mu_1=\mu_\mathrm m+\delta\mu_1(1-q)\pm\delta\mu_2(1-q)^{3/2}+O\big((1-q)^2\big)$
 where $\delta\mu_1$ is given by the coefficient in \eqref{eq:expansion.mu.1}
 and
 \begin{equation}
@@ -1004,7 +1064,7 @@ by the coefficient in \eqref{eq:expansion.E.1} and
           f'(1)(2f''(1)-(2-(2-\delta q_0)\delta q_0)f'''(1))
         \big]
         \\
-    &\hspace{17pc}\times
+    &\hspace{15pc}\times
         \big[f'(1)\big(6f''(1)+(18-(6-\delta q_0)\delta q_0)f'''(1)\big)
         \big]
     \bigg)^\frac12
@@ -1014,22 +1074,9 @@ and $\delta q_0$ is the coefficient in the expansion $q_0=1-\delta q_0(1-q)+O((1
 \begin{equation}
   0=((16-(6-\delta q_0)\delta q_0)\delta q_0-12)f'(1)f'''(1)-2\delta q_0(f''(1)-f'(1))f''(1)
 \end{equation}
-
-\begin{figure}
-  \centering
-  \includegraphics{figs/expansion_energy.pdf}
-  \hspace{1em}
-  \includegraphics{figs/expansion_stability.pdf}
-
-  \caption{
-    Demonstration of the convergence of the $(1-q)$-expansion for marginal
-    reference minima. Solid lines and shaded region show are the same as in
-    Fig.~\ref{fig:marginal.prop.above} for $E_0-E_\mathrm{th}\simeq0.00667$.
-    The dotted lines show the expansion of most common neighbors, while the
-    dashed lines in both plots show the expansion for the minimum and maximum
-    energies and stabilities found at given $q$.
-  } \label{fig:expansion}
-\end{figure}
+These predictions from the small $1-q$ expansion are compared with numeric
+saddle points for the complexity of marginal minima in
+Fig.~\ref{fig:expansion}, and the results agree well at small $1-q$.
 
 
 \section{Isolated eigenvalue}
@@ -1050,58 +1097,6 @@ ground state is located in the direction of the eigenvector associated with the
 smallest eigenvalue, and the ground state energy is proportional to that
 eigenvalue.
 
-Our matrix of interest is the Hessian evaluated at a stationary point of the mixed spherical
-model, conditioned on the relative position, energies, and stabilities
-discussed above. We must restrict the artificial spherical model to lie in the
-tangent plane of the `real' spherical configuration space at the point of
-interest, to avoid our eigenvector pointing in a direction that violates the
-spherical constraint. The free energy of this model given a point $\mathbf s$
-and a specific realization of the disordered Hamiltonian is
-\begin{equation}
-  \begin{aligned}
-    \beta F_H(\beta\mid\mathbf s,\omega)
-    &=-\frac1N\log\left(\int d\mathbf x\,\delta(\mathbf x\cdot\mathbf s)\delta(\|\mathbf x\|^2-N)\exp\left\{
-        -\beta\frac12\mathbf x^T\operatorname{Hess}H(\mathbf s,\omega)\mathbf x
-    \right\}\right) \\
-    &=-\lim_{\ell\to0}\frac1N\frac\partial{\partial\ell}\int\left[\prod_{\alpha=1}^\ell d\mathbf x_\alpha\,\delta(\mathbf x_\alpha^T\mathbf s)\delta(N-\mathbf x_\alpha^T\mathbf x_\alpha)\exp\left\{
-      -\beta\frac12\mathbf x^T_\alpha\big(\partial\partial H(\mathbf s)+\omega I\big)\mathbf x_\alpha
-    \right\}\right]
-  \end{aligned}
-\end{equation}
-where the first $\delta$-function keeps the configurations in the tangent
-plane, and the second enforces the spherical constraint. We have anticipated
-treating the logarithm with replicas. We are interested in points $\mathbf s$
-that have certain properties: they are stationary points of $H$ with given
-energy density and stability, and fixed overlap from a reference configuration
-$\pmb\sigma$. We therefore average the free energy above over such points,
-giving
-\begin{equation}
-  \begin{aligned}
-    F_H(\beta\mid E_1,\mu_1,q,\pmb\sigma)
-    &=\int\frac{d\nu_H(\mathbf s,\omega\mid E_1,\mu_1)\delta(Nq-\pmb\sigma\cdot\mathbf s)}{\int d\nu_H(\mathbf s',\omega'\mid E_1,\mu_1)\delta(Nq-\pmb\sigma\cdot\mathbf s')}F_H(\beta\mid\mathbf s,\omega) \\
-    &=\lim_{n\to0}\int\left[\prod_{a=1}^nd\nu_H(\mathbf s_a,\omega_a\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s_a)\right]F_H(\beta\mid\mathbf s_1,\omega_1)
-  \end{aligned}
-\end{equation}
-again anticipating the use of replicas. Finally, the reference configuration $\pmb\sigma$ should itself be a stationary point of $H$ with its own energy density and stability. Averaging over these conditions gives
-\begin{equation}
-  \begin{aligned}
-    F_H(\beta\mid E_1,\mu_1,E_2,\mu_2,q)
-    &=\int\frac{d\nu_H(\pmb\sigma,\varsigma\mid E_0,\mu_0)}{\int d\nu_H(\pmb\sigma',\varsigma'\mid E_0,\mu_0)}\,F_H(\beta\mid E_1,\mu_1,q,\pmb\sigma) \\
-    &=\lim_{m\to0}\int\left[\prod_{a=1}^m d\nu_H(\pmb\sigma_a,\varsigma_a\mid E_0,\mu_0)\right]\,F_H(\beta\mid E_1,\mu_1,q,\pmb\sigma_1)
-  \end{aligned}
-\end{equation}
-This formidable expression is now ready to be averaged over the disordered Hamiltonians $H$. Once averaged,
-the minimum eigenvalue of the conditioned Hessian is then given by twice the ground state energy, or
-\begin{equation}
-  \lambda_\text{min}=2\lim_{\beta\to\infty}\overline{F_H(\beta\mid E_1,\mu_1,E_2,\mu_2,q)}
-\end{equation}
-For this calculation, there are three different sets of replicated variables.
-Note that, as for the computation of the complexity, the $\pmb\sigma_1$ and
-$\mathbf s_1$ replicas are \emph{special}. The first again is the only of the
-$\sigma$ replicas constrained to lie at fixed overlap with \emph{all} the
-$\mathbf s$ replicas, and the second is the only of the $\mathbf s$ replicas at
-which the Hessian is evaluated.
-
 \begin{figure}
   \centering
   \begin{tikzpicture}
@@ -1161,9 +1156,61 @@ which the Hessian is evaluated.
     constrained to be at fixed overlap $q$ with the first of the $\pmb\sigma$
     replicas, the reference configuration. All of the $\pmb\sigma$ replicas lie
     on the sphere.
-  }
+  } \label{fig:sphere}
 \end{figure}
 
+Our matrix of interest is the Hessian evaluated at a stationary point of the mixed spherical
+model, conditioned on the relative position, energies, and stabilities
+discussed above. We must restrict the artificial spherical model to lie in the
+tangent plane of the `real' spherical configuration space at the point of
+interest, to avoid our eigenvector pointing in a direction that violates the
+spherical constraint. A sketch of the setup is shown in Fig.~\ref{fig:sphere}. The free energy of this model given a point $\mathbf s$
+and a specific realization of the disordered Hamiltonian is
+\begin{equation}
+  \begin{aligned}
+    \beta F_H(\beta\mid\mathbf s,\omega)
+    &=-\frac1N\log\left(\int d\mathbf x\,\delta(\mathbf x\cdot\mathbf s)\delta(\|\mathbf x\|^2-N)\exp\left\{
+        -\beta\frac12\mathbf x^T\operatorname{Hess}H(\mathbf s,\omega)\mathbf x
+    \right\}\right) \\
+    &=-\lim_{\ell\to0}\frac1N\frac\partial{\partial\ell}\int\left[\prod_{\alpha=1}^\ell d\mathbf x_\alpha\,\delta(\mathbf x_\alpha^T\mathbf s)\delta(N-\mathbf x_\alpha^T\mathbf x_\alpha)\exp\left\{
+      -\beta\frac12\mathbf x^T_\alpha\big(\partial\partial H(\mathbf s)+\omega I\big)\mathbf x_\alpha
+    \right\}\right]
+  \end{aligned}
+\end{equation}
+where the first $\delta$-function keeps the configurations in the tangent
+plane, and the second enforces the spherical constraint. We have anticipated
+treating the logarithm with replicas. We are interested in points $\mathbf s$
+that have certain properties: they are stationary points of $H$ with given
+energy density and stability, and fixed overlap from a reference configuration
+$\pmb\sigma$. We therefore average the free energy above over such points,
+giving
+\begin{equation}
+  \begin{aligned}
+    F_H(\beta\mid E_1,\mu_1,q,\pmb\sigma)
+    &=\int\frac{d\nu_H(\mathbf s,\omega\mid E_1,\mu_1)\delta(Nq-\pmb\sigma\cdot\mathbf s)}{\int d\nu_H(\mathbf s',\omega'\mid E_1,\mu_1)\delta(Nq-\pmb\sigma\cdot\mathbf s')}F_H(\beta\mid\mathbf s,\omega) \\
+    &=\lim_{n\to0}\int\left[\prod_{a=1}^nd\nu_H(\mathbf s_a,\omega_a\mid E_1,\mu_1)\,\delta(Nq-\pmb\sigma\cdot\mathbf s_a)\right]F_H(\beta\mid\mathbf s_1,\omega_1)
+  \end{aligned}
+\end{equation}
+again anticipating the use of replicas. Finally, the reference configuration $\pmb\sigma$ should itself be a stationary point of $H$ with its own energy density and stability. Averaging over these conditions gives
+\begin{equation}
+  \begin{aligned}
+    F_H(\beta\mid E_1,\mu_1,E_2,\mu_2,q)
+    &=\int\frac{d\nu_H(\pmb\sigma,\varsigma\mid E_0,\mu_0)}{\int d\nu_H(\pmb\sigma',\varsigma'\mid E_0,\mu_0)}\,F_H(\beta\mid E_1,\mu_1,q,\pmb\sigma) \\
+    &=\lim_{m\to0}\int\left[\prod_{a=1}^m d\nu_H(\pmb\sigma_a,\varsigma_a\mid E_0,\mu_0)\right]\,F_H(\beta\mid E_1,\mu_1,q,\pmb\sigma_1)
+  \end{aligned}
+\end{equation}
+This formidable expression is now ready to be averaged over the disordered Hamiltonians $H$. Once averaged,
+the minimum eigenvalue of the conditioned Hessian is then given by twice the ground state energy, or
+\begin{equation}
+  \lambda_\text{min}=2\lim_{\beta\to\infty}\overline{F_H(\beta\mid E_1,\mu_1,E_2,\mu_2,q)}
+\end{equation}
+For this calculation, there are three different sets of replicated variables.
+Note that, as for the computation of the complexity, the $\pmb\sigma_1$ and
+$\mathbf s_1$ replicas are \emph{special}. The first again is the only of the
+$\sigma$ replicas constrained to lie at fixed overlap with \emph{all} the
+$\mathbf s$ replicas, and the second is the only of the $\mathbf s$ replicas at
+which the Hessian is evaluated.
+
 Using the same methodology as above, the disorder-dependent terms are captured in the linear operator
 \begin{equation}
   \mathcal O(\mathbf t)=
@@ -1408,7 +1455,7 @@ The basic form of the action is (for replica symmetric $A$)
   2\mathcal S_x(\mathcal Q_x\mid\mathcal Q)
   =-\beta\mu+\frac12\beta^2f''(1)(1-a_0^2)+\log(1-a_0)+\frac{a_0}{1-a_0}+\mathcal X^T\left(\beta B-\frac1{1-a_0}C\right)\mathcal X
 \end{equation}
-where the matrix $B$ comes from the $\mathcal X$-dependant parts of the first
+where the matrix $B$ comes from the $\mathcal X$-dependent parts of the first
 lines of \eqref{eq:action.eigenvalue} and is given by
 \begin{equation}
   B=\begin{bmatrix}
@@ -1581,7 +1628,7 @@ Both the denominator and the logarithm are treated using the replica trick, whic
 \end{equation}
 The derivation of this proceeds in much the same way as for the complexity or
 the isolated eigenvalue. Once the $\delta$-functions are converted to
-exponentials, the $H$-dependant terms can be expressed by convolution with the
+exponentials, the $H$-dependent terms can be expressed by convolution with the
 linear operator
 \begin{equation}
   \mathcal O(\mathbf t)