More polishing.

1 files changed, 25 insertions, 14 deletions

diff --git a/2-point.tex b/2-point.tex
index 443602a..8c2107d 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -1207,7 +1207,7 @@ the minimum eigenvalue of the conditioned Hessian is then given by twice the gro
 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
+$\pmb\sigma$ 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.
 
@@ -1221,7 +1221,7 @@ Using the same methodology as above, the disorder-dependent terms are captured i
   \delta(\mathbf t-\mathbf s_1)\beta\sum_c^\ell(\mathbf x_c\cdot\partial_{\mathbf t})^2
 \end{equation}
 that is applied to $H$ by integrating over $\mathbf t\in\mathbb R^N$. The
-resulting expression for the integrand is produces dependencies only on the
+resulting expression for the integrand produces dependencies only on the
 scalar products in \eqref{eq:fields} and on the new scalar products involving
 the tangent plane vectors $\mathbf x$,
 \begin{align}
@@ -1287,7 +1287,15 @@ given by
 \end{equation}
 As usual in these quenched Franz--Parisi style computations, the saddle point expressions for the variables $\mathcal Q$ in the joint limits of $m$, $n$, and $\ell$ to zero are independent of $\mathcal Q_x$, and so these quantities take the same value they do for the two-point complexity that we computed above. The saddle point conditions for the variables $\mathcal Q_x$ are then fixed by extremizing with respect to the final action.
 
-To evaluate this expression, we need a sensible ansatz for the variables $\mathcal Q_x$. The matrix $A$ we expect to be an ordinary hierarchical matrix, and since the model is a spherical 2-spin the finite but low temperature order will be {\oldstylenums1}\textsc{rsb}. The expected form of the $X$ matrices follows our reasoning for the 01 matrices of the previous section: namely, they should have constant rows and a column structure which matches that of the level of \textsc{rsb} order associated with the degrees of freedom that parameterize the columns. Since both the reference configurations and the constrained configurations have replica symmetric order, we expect
+To evaluate this expression, we need a sensible ansatz for the variables
+$\mathcal Q_x$. The matrix $A$ we expect to be an ordinary hierarchical matrix,
+and since the model is a spherical 2-spin the finite but low temperature order
+will be replica symmetric with nonzero $a_0$. The expected form of the $X$ matrices
+follows our reasoning for the 01 matrices of the previous section: namely, they
+should have constant rows and a column structure which matches that of the
+level of \textsc{rsb} order associated with the degrees of freedom that
+parameterize the columns. Since both the reference configurations and the
+constrained configurations have replica symmetric order, we expect
 \begin{align}
   X^0
   =
@@ -1351,7 +1359,7 @@ compile in the vector $\mathcal X=(x_0,\hat x_0,x_1\hat x_1^1,\hat x_1^0)$.
 
 Inserting this ansatz is straightforward in the first part of
 \eqref{eq:action.eigenvalue}, but the term with $\log\det$ is more complicated.
-We must invert the block matrix inside. Defining
+We must invert the block matrix inside. We define
 \begin{equation}
     \begin{bmatrix}
       C^{00}&iR^{00}&C^{01}&iR^{01}\\
@@ -1415,7 +1423,7 @@ where the blocks inside the inverse are given by
       \end{bmatrix}
     \right)^{-1}
 \end{align}
-Here, $M_{22}$ is the inverse of the matrix already analyzed in as part of
+Here, $M_{22}$ is the inverse of the matrix already analyzed as part of
 \eqref{eq:two-point.action}. Following our discussion of the inverses of block
 replica matrices above, and reasoning about their products with the rectangular
 block constant matrices, things can be worked out from here. For instance, the
@@ -1539,7 +1547,7 @@ Note that the second of these conditions implies that the quadratic form in
 $\mathcal X$ in the action vanishes at the saddle.
 
 We would like to take the limit of $\beta\to\infty$. As is usual in the
-two-spin model, the appropriate limits of the order parameter is
+two-spin model, the appropriate limit of the order parameter is
 $a_0=1-(y\beta)^{-1}$. Upon inserting this scaling and taking the limit, we
 finally find
 \begin{equation}
@@ -1642,7 +1650,7 @@ Averaging over $H$ squares the application of this operator to $f$ as before.
 After performing a Hubbard--Stratonovich using matrix order parameters
 identical to those used in the calculation of the complexity, we find that
 \begin{equation}
-  \beta V_\beta(q\mid E_0,\mu_0)=-\frac1N\lim_{\substack{m\to0\\n\to0}}\frac\partial{\partial n}\int d\mathcal Q_0\,d\mathcal Q_1\,e^{Nm\mathcal S_0(\mathcal Q_0)+Nn\mathcal S_\mathrm{FP}(\mathcal Q_1)}
+  \beta V_\beta(q\mid E_0,\mu_0)=-\frac1N\lim_{\substack{m\to0\\n\to0}}\frac\partial{\partial n}\int d\mathcal Q_0\,d\mathcal Q_1\,e^{Nm\mathcal S_0(\mathcal Q_0)+Nn\mathcal S_\mathrm{FP}(\mathcal Q_1\mid\mathcal Q_0)}
 \end{equation}
 where $\mathcal S_0$ is the same as in \eqref{eq:one-point.action} and
 \begin{equation}
@@ -1656,7 +1664,7 @@ where $\mathcal S_0$ is the same as in \eqref{eq:one-point.action} and
     Q-\begin{bmatrix}C^{01}\\iR^{10}\end{bmatrix}^T\begin{bmatrix}C^{00}&iR^{00}\\iR^{00}&D^{00}\end{bmatrix}^{-1}\begin{bmatrix}C^{01}\\iR^{10}\end{bmatrix}
     \right)
 \end{equation}
-Here, because we are at finite temperature for the constrained configuration,
+Here, because we are at low but nonzero temperature for the constrained configuration,
 we make a {\oldstylenums1}\textsc{rsb} anstaz for the matrix $Q$, while the
 $00$ matrices will take their saddle point value for the one-point complexity
 and the $01$ matrices have the same structure as \eqref{eq:01.ansatz}.
@@ -1665,8 +1673,9 @@ Inserting these gives
   \begin{aligned}
     \beta V_\beta&=\frac12\beta^2\big[f(1)-(1-x)f(q_1)-xf(q_0)\big]
     +\beta\hat\beta_0f(q)+\beta r^{10}f'(q)-\frac{1-x}x\log(1-q_1)
-    +\frac1x\log(1-(1-x)q_1-xq_0) \\
-     &+\frac{q_0-d^{00}_df'(1)q^2-2r^{00}_df'(1)r^{10}q+(r^{10})^2f'(1)}{
+    \\
+                 &\qquad+\frac1x\log(1-(1-x)q_1-xq_0)
+     +\frac{q_0-d^{00}_df'(1)q^2-2r^{00}_df'(1)r^{10}q+(r^{10})^2f'(1)}{
       1-(1-x)q_1-xq_0
     }
   \end{aligned}
@@ -1680,9 +1689,11 @@ $q_0=1-(y_0\beta)^{-1}$. Inserting this, the limit is
 \end{equation}
 The saddle point in $y_0$ can now be taken, taking care to choose the solution for $y_0>0$. This gives
 \begin{equation}
-  V_\infty^{\textsc{rs}}=-\hat\beta_0f(q)-r^{11}_\mathrm df'(q)q-\sqrt{(1-q^2)\left(1-\frac{f'(q)^2}{f'(1)^2}\right)}
+  V_\infty^{\textsc{rs}}(q\mid E_0,\mu_0)=-\hat\beta_0f(q)-r^{11}_\mathrm df'(q)q-\sqrt{(1-q^2)\left(1-\frac{f'(q)^2}{f'(1)^2}\right)}
 \end{equation}
-The second case is when the inner statistical mechanics problem has replica symmetry breaking. Here, $q_0$ approaches a nontrivial limit, but $x=z\beta^{-1}$ approaches zero and $q_1=1-(y_1\beta)^{-1}$ approaches one.
+The second case is when the inner statistical mechanics problem has replica
+symmetry breaking. Here, $q_0$ approaches a nontrivial limit, but
+$x=z\beta^{-1}$ approaches zero and $q_1=1-(y_1\beta)^{-1}$ approaches one. The result is
 \begin{equation}
   \begin{aligned}
     V_\infty^{\oldstylenums{1}\textsc{rsb}}(q\mid E_0,\mu_0)
@@ -1708,7 +1719,7 @@ found something striking: only those at the threshold energy have other
 marginal minima nearby. For the many marginal minima away from the threshold
 (including the exponential majority), there is a gap in overlap between them.
 
-This has implications for pictures of dynamical relaxation. In most $p+s$
+This has implications for pictures of relaxation and aging. In most $p+s$
 models studied, quenches from infinite to zero temperature (gradient descent
 starting from a random point) relax towards marginal states with energies above
 the threshold energy \cite{Folena_2023_On}, while at least in some models a
@@ -1717,7 +1728,7 @@ relaxes towards marginal states with energies below the threshold energy
 \cite{Folena_2020_Rethinking, Folena_2021_Gradient}. We found (see especially
 Figs.~\ref{fig:marginal.prop.below} and \ref{fig:marginal.prop.above}) that the
 neighborhoods of marginal states above and below the threshold are quite
-different, and yet the emergent aging behaviors relaxing to states above and
+different, and yet the emergent aging behaviors relaxing toward states above and
 below the threshold seem to be the same. Therefore, this kind of dynamics
 appears to be insensitive to the neighborhood of the marginal state being
 approached. To understand something better about why certain states attract the