More small edits, now in section 3

1 files changed, 29 insertions, 23 deletions

diff --git a/marginal.tex b/marginal.tex
index c1b5a82..190c3be 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -445,10 +445,10 @@ examples in the next section.
 \label{sec:marginal.kac-rice}
 
 The situation in the study of random landscapes is often as follows: an
-ensemble of smooth functions $H:\mathbb R^N\to\mathbb R$ define random
+ensemble of smooth energy functions $H:\mathbb R^N\to\mathbb R$ defines a family of random
 landscapes, often with their configuration space subject to one or more
 constraints of the form $g(\mathbf x)=0$ for $\mathbf x\in\mathbb R^N$. The
-geometry of such landscapes is studied by their complexity, or the average
+typical geometry of landscapes drawn from the ensemble is studied by their complexity, or the average
 logarithm of the number of stationary points with certain properties, e.g., of
 marginal minima at a given energy.
 
@@ -461,7 +461,7 @@ extremizing the Lagrangian
 \end{equation}
 with respect to $\mathbf x$ and the Lagrange multipliers
 $\pmb\omega=\{\omega_1,\ldots,\omega_r\}$. The corresponding gradient and
-Hessian for the problem are
+Hessian of the energy associated with this constrained extremal problem are
 \begin{align}
   &\nabla H(\mathbf x,\pmb\omega)
   =\partial L(\mathbf x,\pmb\omega)
@@ -474,7 +474,9 @@ Hessian for the problem are
   \end{aligned}
 \end{align}
 where $\partial=\frac\partial{\partial\mathbf x}$ will always represent the
-derivative with respect to the vector argument $\mathbf x$. The number of
+derivative with respect to the vector argument $\mathbf x$.
+
+The number of
 stationary points in a landscape for a particular function $H$ is found by
 integrating over the Kac--Rice measure
 \begin{equation} \label{eq:kac-rice.measure}
@@ -488,7 +490,7 @@ integrating over the Kac--Rice measure
 \end{equation}
 with a $\delta$-function of the gradient and the constraints ensuring that we
 count valid stationary points, and the determinant of the Hessian serving as
-the Jacobian of the argument to the $\delta$-function. It is usually more
+the Jacobian of the argument to the $\delta$ function \cite{Kac_1943_On, Rice_1939_The}. It is usually more
 interesting to condition the count on interesting properties of the stationary
 points, like the energy and spectrum trace, or
 \begin{equation} \label{eq:kac-rice.measure.2}
@@ -499,7 +501,7 @@ points, like the energy and spectrum trace, or
     \,\delta\big(N\mu-\operatorname{Tr}\operatorname{Hess}H(\mathbf x,\pmb\omega)\big)
   \end{aligned}
 \end{equation}
-We further want to control the value of the minimum eigenvalue of the Hessian
+We specifically want to control the value of the minimum eigenvalue of the Hessian
 at the stationary points. Using the method introduced in Section
 \ref{sec:eigenvalue}, we can write the number of stationary points with energy
 $E$, Hessian trace $\mu$, and smallest eigenvalue $\lambda^*$ as
@@ -514,24 +516,28 @@ $E$, Hessian trace $\mu$, and smallest eigenvalue $\lambda^*$ as
     \delta\big(N\lambda^*-\mathbf s^T\operatorname{Hess}H(\mathbf x,\pmb\omega)\mathbf s\big)
   \end{aligned}
 \end{equation}
+\end{widetext}
 where the additional $\delta$-functions
 \begin{equation}
   \delta(\mathbf s^T\partial\mathbf g(\mathbf x))
   =\prod_{s=1}^r\delta(\mathbf s^T\partial g_i(\mathbf x))
 \end{equation}
-ensure that the integrals are constrained to the tangent space of the
-configuration manifold at the point $\mathbf x$. The complexity of points with
-a specific energy, stability, and minimum eigenvalue is defined as the average
-over functions $H$ of the logarithm of the number $\mathcal N_H$ of stationary
-points, or
+ensure that the integrals involving potential eigenvectors $\mathbf s$ are constrained to
+the tangent space of the configuration manifold at the point $\mathbf x$.
+
+The
+complexity of points with a specific energy, stability, and minimum eigenvalue
+is defined as the average over the ensemble of functions $H$ of the logarithm
+of the number $\mathcal N_H$ of stationary points, or
 \begin{equation}
   \Sigma_{\lambda^*}(E,\mu)
   =\frac1N\overline{\log\mathcal N_H(E,\mu,\lambda^*)}
 \end{equation}
 In practice, this can be computed by introducing replicas to treat the
-logarithm ($\log x=\lim_{n\to0}\frac\partial{\partial n}x^n$) and replicating
-again to treat each of the normalizations in the numerator
+logarithm ($\log x=\lim_{n\to0}\frac\partial{\partial n}x^n$) and introducing another set of replicas
+to treat each of the normalizations in the numerator
 ($x^{-1}=\lim_{m\to-1}x^m$). This leads to the expression
+\begin{widetext}
 \begin{equation} \label{eq:min.complexity.expanded}
   \begin{aligned}
     \Sigma_{\lambda^*}(E,\mu)
@@ -563,7 +569,7 @@ value of the minimum eigenvalue $\lambda^*$, or
   0=\frac\partial{\partial\lambda^*}\Sigma_{\lambda^*}(E,\mu_\text{m}(E))\bigg|_{\lambda^*=0}
 \end{equation}
 The marginal complexity follows by evaluating the complexity conditioned on
-$\lambda_{\text{min}}=0$ at the marginal stability $\mu=\mu_\text{m}(E)$,
+$\lambda^*=0$ at the marginal stability $\mu=\mu_\text{m}(E)$,
 \begin{equation} \label{eq:marginal.complexity}
   \Sigma_\text{m}(E)
   =\Sigma_0(E,\mu_\text m(E))
@@ -575,7 +581,7 @@ $\lambda_{\text{min}}=0$ at the marginal stability $\mu=\mu_\text{m}(E)$,
 Several elements of the computation of the marginal complexity, and indeed the
 ordinary dominant complexity, follow from the formulae of the above section in
 the same way. The physicists' approach to this problem seeks to convert all of
-the Kac--Rice measure defined in \eqref{eq:kac-rice.measure} and
+the components of the Kac--Rice measure defined in \eqref{eq:kac-rice.measure} and
 \eqref{eq:kac-rice.measure.2} into elements of an exponential integral over
 configuration space. To begin with, all Dirac $\delta$ functions are
 expressed using their Fourier representation, with
@@ -593,29 +599,29 @@ expressed using their Fourier representation, with
   \end{aligned}
 \end{align}
 To do this we have introduced auxiliary fields $\hat{\mathbf x}_a$,
-$\hat\beta_a$, and $\hat\lambda_a$. Since the permutation symmetry of replica vectors
+$\hat\beta_a$, and $\hat\lambda_a$. Because the permutation symmetry of replica vectors
 is preserved in \textsc{rsb} orders, the order parameters $\hat\beta$
 and $\hat\lambda$ will quickly lose their indices, since they will ubiquitously
-be constant over the replicas index at the eventual saddle point solution.
+be constant over the replica index at the eventual saddle point solution.
 
 We would like to make a similar treatment of the determinant of the Hessian
 that appears in \eqref{eq:kac-rice.measure}. The standard approach is to drop
 the absolute value function around the determinant. This can potentially lead
 to severe problems with the complexity. However, it is a justified step when
-the parameters of the problem, i.e., $E$, $\mu$, and $\lambda^*$, put us in a
+the parameters of the problem $E$, $\mu$, and $\lambda^*$ put us in a
 regime where the exponential majority of stationary points have the same index.
 This is true for maxima and minima, and for saddle points whose spectra have a
 strictly positive bulk with a fixed number of negative outliers. It is in
-particular a safe operation for this problem of marginal minima, which lie
+particular a safe operation for the present problem of marginal minima, which lie
 right at the edge of disaster.
 
-Dropping the absolute value sign allows us to write
+Dropping the absolute value function allows us to write
 \begin{equation} \label{eq:determinant}
   \det\operatorname{Hess}H(\mathbf x_a, \pmb\omega_a)
   =\int d\bar{\pmb\eta}_a\,d\pmb\eta_a\,
   e^{-\bar{\pmb\eta}_a^T\operatorname{Hess}H(\mathbf x_a,\pmb\omega_a)\pmb\eta_a}
 \end{equation}
-for $N$-dimensional Grassmann vectors $\bar{\pmb\eta}_a$ and $\pmb\eta_a$. For
+using $N$-dimensional Grassmann vectors $\bar{\pmb\eta}_a$ and $\pmb\eta_a$. For
 the spherical models this step is unnecessary, since there are other ways to
 treat the determinant keeping the absolute value signs, as in previous works
 \cite{Folena_2020_Rethinking, Kent-Dobias_2023_How}. However, other of
@@ -624,7 +630,7 @@ our examples are for models where the same techniques are impossible.
 For the cases studied here, fixing the trace results in a relationship
 between $\mu$ and the Lagrange multipliers enforcing the constraints. This is
 because the trace of $\partial\partial H$ is typically an order of $N$ smaller
-than that of the constraint functions $\partial\partial g_i$. The result is that
+than the trace of $\partial\partial g_i$. The result is that
 \begin{equation}
   \mu
   =\frac1N\operatorname{Tr}\operatorname{Hess}H(\mathbf x)
@@ -1756,7 +1762,7 @@ basis has $2^{2D-1}$ elements. For instance, for $\mathbb R^{N|4}$ we have $\mat
 \label{sec:brst}
 
 When the trace $\mu$ is not fixed, there is an unusual symmetry in the dominant
-complexity of minima \cite{Annibale_2004_Coexistence, Kent-Dobias_2023_How}.
+complexity of minima \cite{Annibale_2003_The, Annibale_2003_Supersymmetric, Annibale_2004_Coexistence}.
 This arises from considering the Kac--Rice formula as a kind of gauge fixing
 procedure \cite{Zinn-Justin_2002_Quantum}. Around each stationary point
 consider making the coordinate transformation $\mathbf u=\nabla H(\mathbf x)$.