Reorganizing.

1 files changed, 17 insertions, 333 deletions

diff --git a/topology.tex b/topology.tex
index 277d8d3..3801eae 100644
--- a/topology.tex
+++ b/topology.tex
@@ -492,6 +492,23 @@ case whenever $V_{\text{\textsc{sat}}\ast}^2<V_\text{on}^2$.
   } \label{fig:phases}
 \end{figure}
 
+\begin{figure}
+  \includegraphics{figs/phases_12_1.pdf}
+  \hspace{-2.85em}
+  \includegraphics{figs/phases_12_2.pdf}
+  \hspace{-2.85em}
+  \includegraphics{figs/phases_12_3.pdf}
+  \hspace{-2.85em}
+  \includegraphics{figs/phases_12_4.pdf}
+
+  \caption{
+    \textbf{Linear--quadratic crossover.}
+    Topological phases for models with a covariance function
+    $f(q)=(1-\lambda)q+\lambda\frac12q^2$ for several values of $\lambda$,
+    interpolating between homogeneous linear ($\lambda=0$) and quadratic ($\lambda=1$) constraints.
+  }
+\end{figure}
+
 The phase diagram implied by these transitions is shown in
 Fig.~\ref{fig:phases} for three different homogeneous $f(q)$. One interesting
 feature occurs in the limit of $\alpha$ to zero. If $V_0$ is likewise rescaled
@@ -525,82 +542,6 @@ S_\chi(m)$ represents breaks down due to the vanishingly small probability of
 finding any stationary points.
 
 
-
-\section{Complexity of a linear test function}
-
-One way to definitely narrow possible interpretations of the average Euler
-characteristic is to compute a complementary average. The Euler characteristic
-is the alternating sum of numbers of critical points of different index. If
-instead we make the direct sum
-\begin{equation} \label{eq:abs.kac-rice}
-  \mathcal N_H(\Omega)=\sum_{i=0}^N\mathcal N_H(\text{index}=i)
-  =\int_\Omega d\mathbf x\,\delta\big(\nabla H(\mathbf x)\big)
-  \,\big|\det\operatorname{Hess}H(\mathbf x)\big|
-\end{equation}
-we find the total number of stationary points. The formula is exactly the same
-as that for the average Euler characteristic except for an absolute value sign
-around the determinant of the Hessian.
-
-Understanding the number of stationary points as a function of latitude $m$
-will clarify the meaning of our effective action for the average Euler
-characteristic in the range of overlaps $m$ where it takes a complex value. This is because the average number of stationary points is a
-nonnegative number. If the region of complex $\mathcal S_\chi$ has a
-well-defined number of stationary points, it indicates that we are looking at a
-situation with a negative average Euler characteristic. In fact, this is what we find below.
-
-To compute the complexity, we follow a similar procedure to the Euler
-characteristic. The main difference lies in how we treat the absolute value
-function around the determinant. Following \cite{Fyodorov_2004_Complexity}, we
-make use of the identity
-\begin{equation} \label{eq:fyodorov.nightmare}
-  \begin{aligned}
-    |\det A|
-  &=\lim_{\epsilon\to0}\frac{(\det A)^2}{\sqrt{\det(A+i\epsilon I)}\sqrt{\det(A-i\epsilon I)}}
-  \\
-  &=\frac1{(2\pi)^N}\int d\bar{\pmb\eta}_1\,d\pmb\eta_1\,d\bar{\pmb\eta}_2\,d\pmb\eta_2\,d\mathbf a_+\,d\mathbf a_2\,
-  e^{
-    -\bar{\pmb\eta}_1^TA\pmb\eta_1-\bar{\pmb\eta}_2^TA\pmb\eta_2
-    -\frac12\mathbf a_+^T(A+i\epsilon I)\mathbf a_+-\frac12\mathbf a_2^T(A-i\epsilon I)\mathbf a_2
-  }
-\end{aligned}
-\end{equation}
-for an $N\times N$ matrix $A$. Here $\bar{\pmb\eta}_1$, $\pmb\eta_1$,
-$\bar{\pmb\eta}_2$, and $\pmb\eta_2$ are Grassmann vectors and $\mathbf a$ and
-$\mathbf b$ are regular vectors. This introduces many new order parameters into
-the problem, but this is a difficulty of scale rather than principle. With this
-identity substituted for the usual determinant one, the problem can be solved
-much as before. The details of this solution are relegated to
-Appendix~\ref{sec:complexity.details}. We again reduce the result to a single integral over the overlap $m$ with the height axis, of the form
-\begin{equation}
-  \overline{\mathcal N_H(\Omega)}
-  \propto\int dm\,e^{N\mathcal S_\mathcal N(m)}
-\end{equation}
-For $m^2>m_c$ the effective action $\mathcal S_\mathcal N(m)$ is the same as
-$\mathcal S_\chi(m)$ for the Euler characteristic, where we have defined
-\begin{equation}
-  m_c^2
-  =1-\frac{2(1-\alpha)f(1)f'(1)}{
-    (2-\alpha)f(1)f''(1)+2(1-\alpha)f'(1)^2
-    -\sqrt{\alpha f''(1)}\sqrt{4V_0^2(1-\alpha)f'(1)^2+\alpha f(1)^2f''(1)}
-  }
-\end{equation}
-For $m^2\leq m_c$ it instead takes the value
-\begin{align}
-  &\mathcal S_\mathcal N^*(m)
-  =
-  -\frac12\alpha\left(
-    1+\frac{V_0^2\big[f''(1)(1-m^2)-f'(1)\big]}{(1-m^2)[f''(1)f(1)+f'(1)^2]-f(1)f'(1)}
-  \right)
-  \\
-  &\quad+\frac12(1-\alpha)\log\left(
-    \frac{f''(1)(1-m^2)}{f'(1)(1-\alpha)}
-  \right)
-  -\frac12\alpha\log\left(
-    \frac{\alpha f'(1)V_0^2}{(1-m^2)[f''(1)f(1)+f'(1)^2]-f(1)f'(1)}
-  \right)
-  \notag
-\end{align}
-
 \section{Implications for the topology of solutions}
 
 It is not straightforward to directly use the average Euler characteristic to
@@ -1284,263 +1225,6 @@ orders in $N$, since for odd-dimensional manifolds the Euler characteristic is
 always zero. When $N+M+1$ is even, we have $\overline{\chi(\Omega)}=2$ to
 leading order in $N$, as specified in the main text.
 
-\section{Details of the calculation of the complexity}
-\label{sec:complexity.details}
-
-Starting from \eqref{eq:abs.kac-rice}, we make the substitution
-\eqref{eq:delta.exp} to treat the Dirac $\delta$-function and use
-\eqref{eq:fyodorov.nightmare} to write the absolute value of the determinant as
-\begin{align}
-  &\big|\det\partial\partial L(\mathbf x,\pmb\omega)\big|
-  =\lim_{\epsilon\to0}\frac1{(2\pi)^{N+M+1}}\int d\bar{\pmb\eta}_1\,d\pmb\eta_1\,d\bar{\pmb\eta}_2\,d\pmb\eta_2\,
-  d\bar{\pmb\gamma}_1\,d\pmb\gamma_1\,d\bar{\pmb\gamma}_2\,d\pmb\gamma_2\,
-  d\mathbf a_1\,d\mathbf a_2\,d\mathbf b_1\,d\mathbf b_2\, \notag \\
-  &\qquad\times\exp\Bigg\{
-    -\frac12\begin{bmatrix}\mathbf a_1^T&\mathbf b_1^T\end{bmatrix}\big(\partial\partial L(\mathbf x,\pmb\omega)+i\epsilon I\big)\begin{bmatrix}\mathbf a_1\\\mathbf b_1\end{bmatrix}
-    -\frac12\begin{bmatrix}\mathbf a_2^T&\mathbf b_2^T\end{bmatrix}\big(\partial\partial L(\mathbf x,\pmb\omega)-i\epsilon I\big)\begin{bmatrix}\mathbf a_2\\\mathbf b_2\end{bmatrix}
-    \notag \\
-  &\hspace{6em}-\begin{bmatrix}\bar{\pmb\eta}_1^T&\bar{\pmb\gamma}_1^T\end{bmatrix}\partial\partial L(\mathbf x,\pmb\omega)\begin{bmatrix}\pmb\eta_1\\\pmb\gamma_1\end{bmatrix}
-    -\begin{bmatrix}\bar{\pmb\eta}_2^T&\bar{\pmb\gamma}_2^T\end{bmatrix}\partial\partial L(\mathbf x,\pmb\omega)\begin{bmatrix}\pmb\eta_2\\\pmb\gamma_2\end{bmatrix}
-  \Bigg\}
-  \label{eq:abs.det.exp}
-\end{align}
-where the $\mathbf a_{\sfrac12}$ are in $\mathbb R^N$, the $\mathbf b_{\sfrac12}$ are in
-$\mathbb R^M$, and the $\pmb\eta_{\sfrac12}$ and $\pmb\gamma_{\sfrac12}$ are Grassmann vectors just as in
-\eqref{eq:det.exp}, except with an extra copy of each. This zoo of vectors
-quickly becomes tiring. Thankfully, there is a way to compactly represent this
-calculation again using superspace vectors.
-
-Consider the vectors
-\begin{align}
-  \pmb\phi(1,2)
-  &=\mathbf x
-  +\bar\theta_1\pmb\eta_1+\bar{\pmb\eta}_1\theta_1\bar\theta_2\theta_2
-  +\bar\theta_2\pmb\eta_2+\bar{\pmb\eta}_2\theta_2\bar\theta_1\theta_1 \\
-  &\qquad+\frac1{\sqrt2}(\bar\theta_1\theta_2+\bar\theta_2\theta_1)\mathbf a_1
-  +\frac i{\sqrt2}(\bar\theta_1\theta_1+\bar\theta_2\theta_2)\mathbf a_2
-  +\bar\theta_1\theta_1\bar\theta_2\theta_2i\hat{\mathbf x}
-  \notag \\
-  \sigma_k(1,2)
-  &=\omega_k
-  +\bar\theta_1\gamma_{1k}+\bar{\gamma}_{1k}\theta_1\bar\theta_2\theta_2
-  +\bar\theta_2\gamma_{2k}+\bar{\gamma}_{2k}\theta_2\bar\theta_1\theta_1 \\
-  &\qquad+\frac1{\sqrt2}(\bar\theta_1\theta_2+\bar\theta_2\theta_1)b_{1k}
-  +\frac i{\sqrt2}(\bar\theta_1\theta_1+\bar\theta_2\theta_2)b_{2k}
-  +\bar\theta_1\theta_1\bar\theta_2\theta_2\hat\omega_k
-  \notag
-\end{align}
-The entire expression for the complexity of stationary points combining
-\eqref{eq:delta.exp} and \eqref{eq:abs.det.exp} can be expressed in the compact
-form
-\begin{equation}
-  \mathcal N_H(\Omega)
-  =\lim_{\epsilon\to0}\int d\pmb\phi\,d\pmb\sigma\,e^{
-   \int d1\,d2\,L(\pmb\phi(1,2),\pmb\sigma(1,2))
-   -\frac{i\epsilon}2
-   (\|\mathbf a_1\|^2-\|\mathbf a_2\|^2+\|\mathbf b_1\|^2-\|\mathbf b_2\|^2)
- }
-\end{equation}
-Note that unlike in the derivation of the average Euler characteristic,
-$\pmb\phi(1,2)$ does not span the entire superspace. Therefore, we will not be
-able to write expressions like the superdeterminant of
-$f(\pmb\phi^T\pmb\phi/N)$, which are formally zero.
-
-Following similar steps as for the Euler characteristic, we first take the average over the random constraint functions $V$. This yields
-\begin{equation}
-  \begin{aligned}
-    \overline{\mathcal N_H(\Omega)}
-    =\int d\pmb\phi\,d\pmb\sigma\,\exp\Bigg\{
-      \frac12\int d1\,d2\,d3\,d4\,\sum_{k=1}^M\sigma_k(1,2)\sigma_k(3,4)f\left(\frac{\pmb\phi(1,2)\cdot\pmb\phi(3,4)}N\right)
-      \\
-      +\int d1\,d2\left[
-        H(\pmb\phi(1,2))
-        +\frac12\sigma_0(1,2)\big(\|\pmb\phi(1,2)\|^2-N\big)
-        -V_0\sum_{k=1}^M\sigma_k(1,2)
-      \right]
-    \Bigg\}
-  \end{aligned}
-\end{equation}
-We once again have a Gaussian integral in the parameters inside the Lagrange
-multipliers $\sigma_k$ for $k=1,\ldots,M$. However, here we must expand the
-superfields at this stage. Before that, we introduce order parameters analogous
-to before. Alongside those in \eqref{eq:ops.bos}, we have
-\begin{align}
-  A_{ij}=\frac{\mathbf a_i\cdot\mathbf a_j}N
-  &&
-  X_i=\frac{\mathbf x\cdot\mathbf a_i}N
-  &&
-  \hat X_i=-i\frac{\hat{\mathbf x}\cdot\mathbf a_i}N
-  &&
-  H_{ij}=\frac{\bar{\pmb\eta}_i\cdot\pmb\eta_j}N
-  &&
-  J=\frac{\pmb\eta_1\cdot\pmb\eta_2}N
-  &&
-  m_i = \frac{\mathbf x_0\cdot\mathbf a_i}N
-\end{align}
-where the $i$ and $j$ run over $1$ and $2$.
-These come with a volume term in the action
-\begin{equation}
-  \begin{aligned}
-    \mathcal V=\frac12\log\det\left(
-      \begin{bmatrix}
-        C   & iR        & X_1       & X_2       \\
-        iR  & D         & i\hat X_1 & i\hat X_2 \\
-        X_1 & i\hat X_1 & A_{11}    & A_{12}    \\
-        X_2 & i\hat X_2 & A_{12}    & A_{22}
-      \end{bmatrix}
-      -\begin{bmatrix}
-        m \\
-        i\hat m \\
-        m_1 \\
-        m_2
-      \end{bmatrix}
-      \begin{bmatrix}
-        m &
-        i\hat m &
-        m_1 &
-        m_2
-      \end{bmatrix}
-    \right) \qquad \\
-    -\frac12\log\det\begin{bmatrix}
-      0 & G_{11} & \bar J & G_{12} \\
-      -G_{11} & 0 & -G_{21} & J \\
-      -\bar J & G_{21} & 0 & G_{22} \\
-      -G_{12} & -J & -G_{22} & 0
-    \end{bmatrix}
-  \end{aligned}
-\end{equation}
-As before, we can immediately integrate out $\sigma_0$, which fixes certain of the order parameters. In particular, we find
-\begin{align}
-  C=1 &&
-  X_1=0 &&
-  X_2=0 &&
-  G_+=-R-\frac12A_+
-\end{align}
-where we have defined the symmetric and antisymmetric combinations of $G_{11}$ and $G_{22}$
-\begin{align}
-  G_+=G_{11}+G_{22}
-  &&
-  G_-=G_{11}-G_{22}
-  &&
-  A_+=A_{11}+A_{22}
-  &&
-  A_-=A_{11}-A_{22}
-\end{align}
-Once the first three substitutions have been made, the result for the Gaussian
-integral in the ordinary variables $\omega_k$, $\hat\omega_k$, $b_{1k}$, and
-$b_{2k}$ is a kernel with
-\begin{equation}
-  K=\begin{bmatrix}
-    K_{11} & iRf'(1) & -\hat X_1 f'(1) & -\hat X_2 f'(1) \\
-    iRf'(1) & f(1) & 0 & 0 \\
-    -\hat X_1 f'(1) & 0 & i\epsilon-\tfrac12f'(1)(A_++A_-) & -A_{12}f'(1) \\
-    -\hat X_2 f'(1) & 0 & -A_{12}f'(1) & -i\epsilon-\tfrac12f'(1)(A_+-A_-)
-  \end{bmatrix}
-\end{equation}
-\begin{equation}
-  K_{11}=Df'(1)-\frac12f''(1)\left[
-      (R-\tfrac12A_+)^2+\tfrac12A_-^2+2A_{12}^2-(G_-^2+4G_{12}G_{21}+4\bar JJ)
-    \right]
-\end{equation}
-and likewise a Gaussian integral in the Grassmann variables with kernel
-\begin{equation}
-  K'=f'(1)\begin{bmatrix}
-    0 & G_{11} & J & G_{21} \\
-    -G_{11} & 0 & -G_{12} & \bar J \\
-    -J & G_{12} & 0 & G_{22} \\
-    -G_{21} & -\bar J & -G_{22} & 0
-  \end{bmatrix}
-\end{equation}
-The effective action then has the form
-\begin{equation}
-  \mathcal S
-  =\hat m+\frac12i\epsilon A_--\frac\alpha2\left(
-    \log\det K-\log\det K'+V_0^2(K^{-1})_{22}
-  \right)+\mathcal V
-\end{equation}
-It is not helpful to write out this entire expression, which is quite large.
-However, we only find saddle points of this action with $\bar
-J=J=G_-=\hat X_1=\hat X_2=m_1=m_2=A_{12}=0$ and $G_{21}=G_{12}$. Setting these variables to zero, we find
-\begin{align}
-  \notag
-  \mathcal S
-  =\hat m
-  -\alpha\frac12\log\frac{
-    f'(1)(Df(1)+R^2f'(1))-(\frac12(R-A_+)^2+A_-^2-2G_{12}^2)f(1)f''(1)
-  }{
-    [\frac12(R+A_+)^2-2G_{12}^2]f'(1)^2
-  }
-  \\
-  \notag
-  +\frac12(1-\alpha)\log\frac{
-    4(A_+^2-A_-^2)
-  }{
-    \frac12(R+A_+)^2-2G_{12}^2
-  }
-  +\frac12\log\frac{D(1-m^2)+R^2-2Rm\hat m+\hat m^2}{
-    \frac12(R+A_+)^2-2G_{12}^2
-  }
-  \\
-  -\frac12\alpha V_0^2\left(
-    f(1)+\frac{R^2f'(1)^2}{Df'(1)-(\frac12(R-A_+)^2+A_-^2-2G_{12}^2)f''(1)}
-  \right)^{-1}
-\end{align}
-One solution to these equations is $A_-=G_{12}=0$ and $A_+=R^*$ with
-$D$, $\hat m$, and $R$ exactly as for the Euler characteristic. The resulting effective
-action as a function of $m$ is also exactly the same. There is another solution, this time with $A_+=R$, $A_-=G_{12}$, and
-\begin{align}
-  &R=\frac{\alpha mf'(1)}{b^2}\big[
-    V_0^2f'(1)^2(1-m^2)-f(1)b
-  \big]
-  \\
-  &G_{12}^2
-  =\frac{f'(1)mR}{f''(1)(1-m^2)b^2}
-  \big[\alpha V_0^2f'(1)^2f''(1)(1-m^2)^2-b^2-\alpha bf'(1)(f(1)-(1-m^2)f'(1))
-  \big]
-\end{align}
-where we have defined the constant
-\begin{equation}
-  b=(1-m^2)[f''(1)f(1)+f'(1)^2]-f(1)f'(1)
-\end{equation}
-The resulting form for the action is
-\begin{align}
-  \mathcal S_\mathcal N(m)
-  =\frac12(1-\alpha)\log\left(
-    \frac{f''(1)(1-m^2)}{f'(1)(1-\alpha)}
-  \right)
-  -\frac12\alpha\log\left(
-    \frac{\alpha f'(1)V_0^2}b
-  \right)
-  -\frac12\alpha\frac{V_0^2\big[f''(1)(1-m^2)-f'(1)\big]+b}b
-\end{align}
-This solution is plotted alongside the solution that coincides with that of the
-Euler characteristic in Fig. It is clear from this plot that the new solution
-cannot be valid in the entire range of $m$, since it diverges as $m$ goes to 1
-where we know there are vanishingly few stationary points. However, there is a
-single point $m_c$ where the two solutions coincide, and they have the
-possibility of trading stability. This is given by
-\begin{equation}
-  m_c^2
-  =1-\frac{2(1-\alpha)f(1)f'(1)}{
-    (2-\alpha)f(1)f''(1)+2(1-\alpha)f'(1)^2
-    -\sqrt{\alpha f''(1)}\sqrt{4V_0^2(1-\alpha)f'(1)^2+\alpha f(1)^2f''(1)}
-  }
-\end{equation}
-
-\begin{equation}
-  \mathcal S_\mathcal N(m)
-  =\begin{cases}
-    \mathcal S_\chi(m) & m^2\geq m_c^2 \\
-    \mathcal S_{\ast}(m) & m^2<m_c^2
-  \end{cases}
-\end{equation}
-This formula remains valid also in the regime when $m_c^2<0$, when the
-complexity is given by the function $\mathcal S_\chi$ in the entire range
-$m\in(-1,1)$.
-
-\begin{equation}
-  V_\text{on}^2=\frac{(1-\alpha)f'(1)^2-\alpha f(1)f''(1)}{\alpha f''(1)}
-\end{equation}
 
 \section{The quenched shattering energy}
 \label{sec:1frsb}