From fb46fd3b95e113ce34d66a0b0bca3380320d9421 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 7 Jul 2022 18:30:04 +0200
Subject: Started coverting all vectors to bold, and general clean-up.

---
 frsb_kac-rice.tex | 190 ++++++++++++++++++++++++++++++++----------------------
 1 file changed, 114 insertions(+), 76 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index ec2b26f..e918176 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -40,40 +40,50 @@ To understand the importance of this computation, consider the following situati
 a transition at a given temperature to a one-step one step symmetry breaking (1RSB) phase at a Kauzmann temperature,
 and, at a lower temperature,
 another  transition to  a full RSB phase (see \cite{gross1985mean,gardner1985spin}, the `Gardner ' phase \cite{charbonneau2014fractal}. 
-Now, this transition involves the lowest, equilibrium states. Because they are obviously unreachable at any reasonable timescale, an often addressed question  to ask is "what is the Gardner transition line for higher than equilibrium energy-densities"? (see, for a review \cite{berthier2019gardner})  For example, when studying `jamming' at zero temperature, the question is posed as to "on what side of the 1RSB-FRS transition
-are the high energy (or low density) states reachable dynamically. 
-Posed in this way, such a question does not have a clear definition.
-In the present paper we give a concrete strategy to define unambiguously such an issue: we consider the local energy minima at a given energy and study their number and other properties: the solution involves a replica-symmetry breaking scheme that is well-defined, and corresponds directly to the topological characteristics of those minima.
+Now, this transition involves the lowest, equilibrium states. Because they are
+obviously unreachable at any reasonable timescale, an often addressed question
+to ask is: what is the Gardner transition line for higher than equilibrium
+energy-densities? (see, for a review \cite{berthier2019gardner})  For example,
+when studying `jamming' at zero temperature, the question is posed as to "on
+what side of the 1RSB-FRS transition are the high energy (or low density)
+states reachable dynamically.  Posed in this way, such a question does not have
+a clear definition.  In the present paper we give a concrete strategy to define
+unambiguously such an issue: we consider the local energy minima at a given
+energy and study their number and other properties: the solution involves a
+replica-symmetry breaking scheme that is well-defined, and corresponds directly
+to the topological characteristics of those minima.
 
 
 
 \section{The model}
 
-Here we consider, for definiteness, the  mixed $p$-spin model, 
+Here we consider, for definiteness, the  mixed $p$-spin model, whose Hamiltonian
 \begin{equation}
-  H(s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}J^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
+  H(\mathbf s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}^NJ^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
 \end{equation}
-for $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$ Then
+is defined for vectors $\mathbf s\in\mathbb R^N$ confined to the sphere $\|\mathbf s\|^2=N$.
+The coupling coefficients are taken at random, with zero mean and covariance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$. This implies that the covariance of the energy with itself depends only on the dot product, or overlap, between two configurations, and in particular that
 \begin{equation}
-  \overline{H(s_1)H(s_2)}=Nf\left(\frac{s_1\cdot 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}
 for
 \begin{equation}
   f(q)=\frac12\sum_pa_pq^p
-\end{equation} 
-or, more generally, the `Toy Model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds} which involve non-polynomial forms for $f(q)$.
+\end{equation}
+More generally, one does not need to start with a Hamiltonian like ours and can simply invoke the covariance rule for arbitrary, non-polynomial $f$, as in the `toy model' of M\'ezard and Parisi \cite{Mezard_1992_Manifolds}.
 
 These may be thought of as a model of generic Gaussian functions on the sphere.
 To constrain the model to the sphere, we use a Lagrange multiplier $\mu$, with the total energy being
 \begin{equation}
-  H(s)+\frac\mu2(s\cdot s-N)
+  H(\mathbf s)+\frac\mu2(\|\mathbf s\|^2-N)
 \end{equation}
 
 At any critical point, the gradient and Hessian are
 \begin{align}
-  \nabla H=\partial H+\mu s &&
-  \operatorname{Hess}H=\partial\partial H+\mu I
+  \nabla H(\mathbf s,\mu)=\partial H(\mathbf s)+\mu\mathbf s &&
+  \operatorname{Hess}H(\mathbf s,\mu)=\partial\partial H(\mathbf s)+\mu I
 \end{align}
+where $\partial=\frac\partial{\partial\mathbf s}$ always.
 The important observation was made by Bray and Dean  \cite{Bray_2007_Statistics} that gradient and Hessian
 are independent for  random Gaussian disorder.
 The average over disorder
@@ -184,8 +194,8 @@ to characterize the saddles, as we shall see later
 \begin{equation}
   \begin{aligned}
     \mathcal N(E, \mu^*)
-    &=\int ds\, d\mu\,\delta\big(\tfrac12(\|s\|^2-N)\big)\,\delta\big(\nabla H(s,\mu)\big)\,\big|\det\operatorname{Hess}H(s,\mu)\big| \\
-    &\hspace{10pc}\times\delta\big(NE-H(s)\big)\delta\big(N\mu^*-\operatorname{Tr}\operatorname{Hess}H(s,\mu)\big)
+    &=\int d\mathbf s\, d\mu\,\delta\big(\tfrac12(\|\mathbf s\|^2-N)\big)\,\delta\big(\nabla H(\mathbf s,\mu)\big)\,\big|\det\operatorname{Hess}H(\mathbf s,\mu)\big| \\
+    &\hspace{10pc}\times\delta\big(NE-H(\mathbf s)\big)\delta\big(N\mu^*-\operatorname{Tr}\operatorname{Hess}H(\mathbf s,\mu)\big)
   \end{aligned}
 \end{equation}
 This number will typically be exponential in $N$. In order to find typical
@@ -239,9 +249,9 @@ In order to average the complexity over disorder properly, the logarithm must be
   \begin{aligned}
     \log\mathcal N(E,\mu^*)
     &=\lim_{n\to0}\frac\partial{\partial n}\mathcal N^n(E,\mu^*) \\
-    &=\lim_{n\to0}\frac\partial{\partial n}\int\prod_a^n ds_a\,d\mu_a\,
-    \delta\big(\tfrac12(\|s_a\|^2-N)\big)\,\delta\big(\nabla H(s_a,\mu_a)\big)\,\big|\det\operatorname{Hess}H(s_a,\mu_a)\big| \\
-    &\hspace{13pc}  \times\delta\big(NE-H(s_a)\big)\delta\big(N\mu^*-\operatorname{Tr}\operatorname{Hess}H(s_a,\mu_a)\big)
+    &=\lim_{n\to0}\frac\partial{\partial n}\int\prod_a^n d\mathbf s_a\,d\mu_a\,
+    \delta\big(\tfrac12(\|\mathbf s_a\|^2-N)\big)\,\delta\big(\nabla H(\mathbf s_a,\mu_a)\big)\,\big|\det\operatorname{Hess}H(\mathbf s_a,\mu_a)\big| \\
+    &\hspace{13pc}  \times\delta\big(NE-H(\mathbf s_a)\big)\delta\big(N\mu^*-\operatorname{Tr}\operatorname{Hess}H(\mathbf s_a,\mu_a)\big)
   \end{aligned}
 \end{equation}
 The replicated Kac--Rice formula was introduced by Ros et al.~\cite{Ros_2019_Complex}, and its
@@ -252,18 +262,19 @@ therefore able to write
 \begin{equation}
   \begin{aligned}
     \Sigma(E, \mu^*)
-    &=\lim_{N\to\infty}\frac1N\lim_{n\to0}\frac\partial{\partial n}\int\left(\prod_a^nds_a\,d\mu_a\right)\,
-    \overline{\prod_a^n \delta\big(\tfrac12(\|s_a\|^2-N)\big)\,\delta\big(\nabla H(s_a,\mu_a)\big)\delta(NE-H(s_a))}\\
+    &=\lim_{N\to\infty}\frac1N\lim_{n\to0}\frac\partial{\partial n}\int\left(\prod_a^nd\mathbf s_a\,d\mu_a\right)\,
+    \overline{\prod_a^n \delta\big(\tfrac12(\|\mathbf s_a\|^2-N)\big)\,\delta\big(\nabla H(\mathbf s_a,\mu_a)\big)\delta(NE-H(\mathbf s_a))}\\
     &\hspace{10pc}
     \times
-    \overline{\prod_a^n |\det\operatorname{Hess}(s_a,\mu_a)|\,\delta\big(N\mu^*-\operatorname{Tr}\operatorname{Hess}H(s_a,\mu_a)\big)}
+    \overline{\prod_a^n |\det\operatorname{Hess}(\mathbf s_a,\mu_a)|\,\delta\big(N\mu^*-\operatorname{Tr}\operatorname{Hess}H(\mathbf s_a,\mu_a)\big)}
   \end{aligned}
 \end{equation}
 
 \subsubsection{The Hessian factors}
 
-The spectrum of the Hessian matrix $\partial\partial H$ is uncorrelated from the gradient. In the large $N$ limit 
-for almost every point and realization of disorder a GOE matrix with variance
+The spectrum of the matrix $\partial\partial H$ is uncorrelated from the
+gradient. In the large $N$ limit for almost every point and realization of
+disorder a GOE matrix with variance
 \begin{equation}
   \overline{(\partial_i\partial_jH)^2}=\frac1Nf''(1)\delta_{ij}
 \end{equation}
@@ -290,7 +301,7 @@ points are minima whose sloppiest eigenvalue is $\mu-\mu_m$.
 To largest order in $N$, the average over the product of determinants factorizes into the product of averages, each of which is given by the same expression depending only on $\mu$:
 \begin{equation}
   \begin{aligned}
-    \overline{\prod_a^n |\det\operatorname{Hess}(s_a,\mu_a)|\,\delta\big(N\mu^*-\operatorname{Tr}\operatorname{Hess}H(s_a,\mu_a)\big)}
+    \overline{\prod_a^n |\det\operatorname{Hess}(\mathbf s_a,\mu_a)|\,\delta\big(N\mu^*-\operatorname{Tr}\operatorname{Hess}H(\mathbf s_a,\mu_a)\big)}
    \rightarrow e^{nN{\cal D}(\mu^*)}\prod_a^n\delta(\mu_a-\mu^*)
   \end{aligned}
 \end{equation}
@@ -310,54 +321,56 @@ with
 
 What we have described is the {\em typical} spectrum for given $\mu$. What about the deviations of the spectrum -- we are particularly interested in the number of negative eigenvalues -- at given $\mu$. The result is well known qualitatively: there are two possibilities:\\
 $\bullet$ For $|\mu|>\mu_m$ there is the possibility of a finite number of eigenvalues of
-the second derivative matrix 
+the second derivative matrix
 
 \subsubsection{The gradient factors}
 
-The $\delta$-functions are treated by writing them in the Fourier basis, introducing auxiliary fields $\hat s_a$ and $\hat\beta$,
+The $\delta$-functions are treated by writing them in the Fourier basis, introducing auxiliary fields $\hat{\mathbf s}_a$ and $\hat\beta$,
 \begin{equation}
-  \prod_a^n \delta\big(\tfrac12(\|s_a\|^2-N)\big)\,\delta\big(\nabla H(s_a,\mu^*)\big)\delta(NE-H(s_a))
-  =\int\frac{d\hat\mu}{2\pi}\,\frac{d\hat\beta}{2\pi}\prod_a^n\frac{d\hat s_a}{2\pi}
-  e^{\frac12\hat\mu(\|s_a\|^2-N)+\hat\beta(NE-H(s_a))+i\hat s_a\cdot(\partial H(s_a)+\mu^*s_a)}
+  \delta\big(\tfrac12(\|\mathbf s_a\|^2-N)\big)\,\delta\big(\nabla H(\mathbf s_a,\mu^*)\big)\delta(NE-H(\mathbf s_a))
+  =\int\frac{d\hat\mu}{2\pi}\,\frac{d\hat\beta}{2\pi}\,\frac{d\hat{\mathbf s}_a}{2\pi}
+  e^{\frac12\hat\mu(\|\mathbf s_a\|^2-N)+\hat\beta(NE-H(\mathbf s_a))+i\hat{\mathbf s}_a\cdot(\partial H(\mathbf s_a)+\mu^*\mathbf s_a)}
 \end{equation}
 $\hat \beta$ is a parameter conjugate to the state energies, i.e. playing the
-role of an inverse temperature for the metastable states. The average over disorder can now be taken, and since everything is Gaussian it gives
+role of an inverse temperature for the metastable states. The average over
+disorder can now be taken for the pieces which depend on $H$, and since
+everything is Gaussian it gives
 \begin{equation}
   \begin{aligned}
     \overline{
       \exp\left\{
-        \sum_a^n(i\hat s_a\cdot\partial_a-\hat\beta)H(s_a)
+        \sum_a^n(i\hat {\mathbf s}_a\cdot\partial_a-\hat\beta)H(s_a)
       \right\}
     }
     &=\exp\left\{
         \frac12\sum_{ab}^n
-        (i\hat s_a\cdot\partial_a-\hat\beta)
-        (i\hat s_b\cdot\partial_b-\hat\beta)
-        \overline{H(s_a)H(s_b)}
+        (i\hat{\mathbf s}_a\cdot\partial_a-\hat\beta)
+        (i\hat{\mathbf s}_b\cdot\partial_b-\hat\beta)
+        \overline{H(\mathbf s_a)H(\mathbf s_b)}
       \right\} \\
     &=\exp\left\{
         \frac N2\sum_{ab}^n
-        (i\hat s_a\cdot\partial_a-\hat\beta)
-        (i\hat s_b\cdot\partial_b-\hat\beta)
-        f\left(\frac{s_a\cdot s_b}N\right)
+        (i\hat{\mathbf s}_a\cdot\partial_a-\hat\beta)
+        (i\hat{\mathbf s}_b\cdot\partial_b-\hat\beta)
+        f\left(\frac{\mathbf s_a\cdot\mathbf s_b}N\right)
       \right\} \\
     &\hspace{-14em}=\exp\left\{
         \frac N2\sum_{ab}^n
         \left[
-          \hat\beta^2f\left(\frac{s_a\cdot s_b}N\right)
-          -2i\hat\beta\frac{\hat s_a\cdot s_b}Nf'\left(\frac{s_a\cdot s_b}N\right)
-          -\frac{\hat s_a\cdot \hat s_b}Nf'\left(\frac{s_a\cdot s_b}N\right)
-          +\left(i\frac{\hat s_a\cdot s_b}N\right)^2f''\left(\frac{s_a\cdot s_b}N\right)
+          \hat\beta^2f\left(\frac{{\mathbf s}_a\cdot {\mathbf s}_b}N\right)
+          -2i\hat\beta\frac{\hat {\mathbf s}_a\cdot {\mathbf s}_b}Nf'\left(\frac{{\mathbf s}_a\cdot {\mathbf s}_b}N\right)
+          -\frac{\hat {\mathbf s}_a\cdot \hat {\mathbf s}_b}Nf'\left(\frac{{\mathbf s}_a\cdot {\mathbf s}_b}N\right)
+          +\left(i\frac{\hat {\mathbf s}_a\cdot {\mathbf s}_b}N\right)^2f''\left(\frac{{\mathbf s}_a\cdot {\mathbf s}_b}N\right)
         \right]
       \right\}
   \end{aligned}
 \end{equation}
 
-We introduce new fields
+We introduce new matrix fields
 \begin{align} \label{eq:fields}
-  C_{ab}=\frac1Ns_a\cdot s_b &&
-  R_{ab}=-i\frac1N\hat s_a\cdot s_b &&
-  D_{ab}=\frac1N\hat s_a\cdot\hat s_b
+  C_{ab}=\frac1N\mathbf s_a\cdot\mathbf s_b &&
+  R_{ab}=-i\frac1N\hat{\mathbf s}_a\cdot{\mathbf s}_b &&
+  D_{ab}=\frac1N\hat{\mathbf s}_a\cdot\hat{\mathbf s}_b
 \end{align}
 Their physical meaning is explained in \S\ref{sec:interpretation}.
 By substituting these parameters into the expressions above and then making a
@@ -643,6 +656,7 @@ As we will see, stable minima are numerous at energies above the ground state,
 but these vanish at the ground state.
 
 \subsection{Expansion near the transition}
+\label{subsec:expansion}
 
 Working with the general equations in their continuum form away from the
 supersymmetric solution is not generally tractable. However, there is another
@@ -690,16 +704,34 @@ Likewise, \eqref{eq:extremum.d} depends linearly on $\bar d$ to all orders, and
       \right\}x_\mathrm{max}+O(x_\mathrm{max}^2)
   \end{aligned}
 \end{equation}
-Finally, the equations for $\bar c$ at first order imply that either $\bar c$
-vanishes as $x_\mathrm{max}$ to zero, or that the linear coefficient vanishes
-as $x_\mathrm{max}$ to zero. Using $\hat\mu$, $\hat\beta$, $r_d$, and $d_d$
-from the annealed solution, this coefficient vanishes when
+The equations cannot be used to find the value of $\bar c$ without going to
+higher order in $x_\mathrm{max}$, but the transition line can be determined by
+examining the stability of the replica symmetric complexity. First, we expand the full form for the complexity about small $x_\textrm{max}$ in the same way as we expand the extremal conditions. To quadratic order, this gives
+\begin{equation}
+  \begin{aligned}
+  \Sigma(E,\mu^*)
+  =\mathcal D(\mu^*)+\hat\beta E-\mu r_d+\frac12\left[\hat\beta^2f(1)+(2\hat\beta r_d-d_d)f'(1)+r_d^2f''(1)\right]+\frac12\log(d_d+r_d^2) \\
+  -\frac12\left[
+    \frac12\hat\beta^2\bar c^2f''(0)+(2\hat\beta\bar r-\bar d)\bar cf''(0)+\bar r^2f''(0)
+    -\frac{\bar d^2-2d_d\bar r^2+d_d^2\bar c^2+4r_d\bar r(\bar d+d_d\bar c)-2r_d^2(\bar c\bar d+\bar r^2)}{2(d_d+r_d^2)^2}
+  \right]x_\textrm{max}^2
+  \end{aligned}
+\end{equation}
+The spectrum Hessian of this expression evaluated at $x_\text{max}=0$ gives the
+stability of the replica symmetric solution with respect to perturbations of
+the type described above. When the values of $\bar r$ and $\bar d$ above are substituted in and everything is evaluated at the replica symmetric solution, the  eigenvalue of interest takes the form
 \begin{equation}
-  \mu^*
+  \lambda
+  =-\bar c^2\frac{(f'(1)-2f(1))^2(f'(1)-f''(0))f''(0)}{2(f'(1)+f''(0))(f'(1)^2-f(1)(f'(1)+f''(1)))^2}(\mu^*-\mu^*_+(E))(\mu^*-\mu^*_-(E))
+\end{equation}
+where
+\begin{equation}
+  \mu^*_\pm(E)
   =\pm\frac{(f'(1)+f''(0))(f'(1)^2-f(1)(f'(1)+f''(1)))}{(2f(1)-f'(1))f'(1)f''(0)^{-1/2}}
-  -\frac{f''(1)-f'(1)}{f'(1)-2f(1)}\epsilon
+  -\frac{f''(1)-f'(1)}{f'(1)-2f(1)}E
 \end{equation}
-We expect that this is the line of stability for the replica symmetric saddle.
+This eigenvalues changes sign when $\mu^*$ crosses $\mu^*_\pm(E)$.  We expect
+that this is the line of stability for the replica symmetric saddle.
 
 
 \section{General solution: examples}
@@ -866,28 +898,30 @@ for different energies and typical vs minima.
 Let $\langle A\rangle$ be the average of $A$ over stationary points with given $E$ and $\mu^*$, i.e.,
 \begin{equation}
   \langle A\rangle
-  =\frac1{\mathcal N}\sum_{\sigma}A(s_\sigma)
+  =\frac1{\mathcal N}\sum_{\sigma}A(\mathbf s_\sigma)
   =\frac1{\mathcal N}
-  \int d\nu(s)\,A(s)
+  \int d\nu(\mathbf s)\,A(\mathbf s)
 \end{equation}
 with
 \begin{equation}
-  d\nu(s)=ds\,d\mu\,\delta\big(\tfrac12(\|s\|^2-N)\big)\,\delta\big(\nabla H(s,\mu)\big)\,\big|\det\operatorname{Hess}H(s,\mu)\big|
-    \delta\big(NE-H(s)\big)\delta\big(N\mu^*-\operatorname{Tr}\operatorname{Hess}H(s,\mu)\big)
+  d\nu(\mathbf s)=d\mathbf s\,d\mu\,\delta\big(\tfrac12(\|\mathbf s\|^2-N)\big)\,\delta\big(\nabla H(\mathbf s,\mu)\big)\,\big|\det\operatorname{Hess}H(\mathbf s,\mu)\big|
+    \delta\big(NE-H(\mathbf s)\big)\delta\big(N\mu^*-\operatorname{Tr}\operatorname{Hess}H(\mathbf s,\mu)\big)
 \end{equation}
-the Kac--Rice measure. The fields $C$, $R$, and $D$ defined in
-\eqref{eq:fields} can be related to certain averages of this type.
+the Kac--Rice measure. Note that this definition of the angle brackets is not
+the same as that used in \S\ref{subsec:expansion} The fields $C$, $R$, and $D$
+defined in \eqref{eq:fields} can be related to certain averages of this type.
 
 \subsection{\textit{C}: distribution of overlaps}
 
 First,
 consider $C$, which has an interpretation nearly identical to that of Parisi's
 $Q$ matrix of overlaps. It can be shown that its off-diagonal corresponds to
-the probability distribution of the overlaps between stationary points $P(q)$. First, define this distribution as
+the probability distribution of the overlaps between stationary points $P(q)$.
+Let $\mathcal S$ be the set of all stationary points with given energy density
+and index. Then
 \begin{equation}
-  P(q)=\frac1{\mathcal N^2}\sum_{\sigma,\sigma'}\delta\left(\frac{s_\sigma\cdot s_{\sigma'}}N-q\right)
+P(q)=\frac1{\mathcal N^2}\sum_{\mathbf s_1\in\mathcal S}\sum_{\mathbf s_2\in\mathcal S}\delta\left(\frac{\mathbf s_1\cdot\mathbf s_2}N-q\right)
 \end{equation}
-where the sum is twice over stationary points $\sigma$ and $\sigma'$. 
 {\em This is the probability that two stationary points randomly drawn from the ensemble
 of stationary points happen to be at overlap $q$}
 
@@ -896,22 +930,26 @@ straightforward to show that moments of this distribution are related to
 certain averages of the form. These are evaluated for a given energy, index, etc, but
 we shall omit these subindices for simplicity.
 
-\begin{eqnarray}
-q^{(p)}
-  &\equiv& \frac1{N^p}\sum_{i_1\cdots i_p}\langle s^1_{i_1}\cdots s^1_{i_p}\rangle\langle s^2_{i_1}\cdots s^2_{i_p}\rangle
-  \nonumber \\
-  &=&\frac1{N^p} \; \frac{1}{{\cal{N}}^2} \left\{ \sum_{{\mathbf s}^1,{\mathbf s}^2}\; \sum_{i_1\cdots i_p} s^1_{i_1}\cdots s^1_{i_p} s^2_{i_1}\cdots s^2_{i_p}\right\}
-  =\frac1{N^p}  \lim_{n\to0} \left\{ \sum_{{\mathbf s}^1,{\mathbf s}^2...{\mathbf s}^n}\; \sum_{i_1\cdots i_p} s^1_{i_1}\cdots s^1_{i_p} s^2_{i_1}\cdots s^2_{i_p}\right\}
-\end{eqnarray}
+\begin{equation}
+  \begin{aligned}
+    q^{(p)}
+    &\equiv \frac1{N^p}\sum_{i_1\cdots i_p}\langle s_{i_1}\cdots s_{i_p}\rangle\langle s_{i_1}\cdots s_{i_p}\rangle
+    =\frac1{N^p} \; \frac{1}{{\cal{N}}^2} \left\{ \sum_{{\mathbf s}_1,{\mathbf s}_2}\; \sum_{i_1\cdots i_p} s^1_{i_1}\cdots s^1_{i_p} s^2_{i_1}\cdots s^2_{i_p}\right\} \\
+    &=\frac1{N^p} \; \frac{1}{{\cal{N}}^2} \left\{\sum_{{\mathbf s}_1,{\mathbf s}_2} (\mathbf s_1\cdot\mathbf s_2)^p\right\}
+    =\frac1{N^p} \lim_{n\to0} \left\{\sum_{{\mathbf s}_1,{\mathbf s}_2,\ldots,\mathbf s_n} (\mathbf s_1\cdot\mathbf s_2)^p\right\}
+  \end{aligned}
+\end{equation}
 The $(n-2)$ extra replicas providing the normalization.
-The average over disorder, and again, for given enegrgy, index,etc reads:
-\begin{eqnarray}
-    \overline{q^{(p)}} &=&\overline{\frac1{N^p}\sum_{i_1\cdots i_p}\langle s_{i_1}\cdots s_{i_p}\rangle\langle s_{i_1}\cdots s_{i_p}\rangle}
-    =\lim_{n\to0}{\int\prod_\alpha^n d\nu(s_\alpha)\,\left(\frac{s_1 s_2}N\right)^p} \times  \Big[ {\mbox {REPLICATED AVERAGED MEASURE}}  \Big]\nonumber \\
-    &=&\lim_{n\to0}{\int D[C,R,D] \,
-    \left(C_{12}\right)^p\; } e^{N\Sigma[C,R,D]}
-    =\frac1{n(n-1)}\lim_{n\to0}{\int D[C,R,D] \,\sum_{a\neq b}\left(C_{ab}\right)^p\; } e^{N\Sigma[C,R,D]}
-\end{eqnarray}
+Replacing the sums over stationary points with integrals over the Kac--Rice measure, the average over disorder, and again, for given energy and index, gives
+\begin{equation}
+  \begin{aligned}
+    \overline{q^{(p)}} &=\overline{\frac1{N^p}\sum_{i_1\cdots i_p}\langle s_{i_1}\cdots s_{i_p}\rangle\langle s_{i_1}\cdots s_{i_p}\rangle}
+    =\lim_{n\to0}{\int\overline{\prod_a^n d\nu(\mathbf s_a)}\,\left(\frac{\mathbf s_1\cdot\mathbf s_2}N\right)^p} \\
+    &=\lim_{n\to0}{\int D[C,R,D] \,
+    \left(C_{12}\right)^p\; } e^{nN\Sigma[C,R,D]}
+    =\frac1{n(n-1)}\lim_{n\to0}{\int D[C,R,D] \,\sum_{a\neq b}\left(C_{ab}\right)^p\; } e^{nN\Sigma[C,R,D]}
+  \end{aligned}
+\end{equation}
 In the last line, we have used that there is nothing special about replicas one and two.
 Using the Parisi ansatz, evaluating by saddle point  {\em summing over all the $n(n-1)$ saddles related by permutation} we then have
 \begin{equation}
-- 
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