From 4e0fc450da1c3b293f232e218cd6d87717e7adb5 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Wed, 4 Sep 2024 16:59:02 +0200
Subject: Begin working on the complexity appendix.

---
 topology.tex | 184 +++++++++++++++++++++++++++++++++++++++++++++++++++++------
 1 file changed, 166 insertions(+), 18 deletions(-)

(limited to 'topology.tex')

diff --git a/topology.tex b/topology.tex
index 9d0428a..4120cef 100644
--- a/topology.tex
+++ b/topology.tex
@@ -417,7 +417,7 @@ 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}
+\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|
@@ -443,15 +443,15 @@ 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}
+\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 b\,
+  &=\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_-\,
   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 b^T(A-i\epsilon I)\mathbf b
+    -\frac12\mathbf a_+^T(A+i\epsilon I)\mathbf a_+-\frac12\mathbf a_-^T(A-i\epsilon I)\mathbf a_-
   }
 \end{aligned}
 \end{equation}
@@ -783,7 +783,7 @@ These new variables can replace $\pmb\phi$ in the integral using a generalized H
 where we show the asymptotic value of the prefactor in Appendix~\ref{sec:prefactor}.
 To move on from this expression,
 we need to expand the superspace notation. We can write
-\begin{equation}
+\begin{equation} \label{eq:ops.q}
   \begin{aligned}
     \mathbb Q(1,2)
     &=C-R(\bar\theta_1\theta_1+\bar\theta_2\theta_2)
@@ -800,7 +800,7 @@ and
 \begin{equation}
   \mathbb M(1)
   =m+\bar\theta_1H_0+\bar H_0\theta_1+i\hat m\bar\theta_1\theta_1
-  \label{eq:ops}
+  \label{eq:ops.m}
 \end{equation}
 with associated measures
 \begin{align}
@@ -811,10 +811,10 @@ with associated measures
   \label{eq:op.measures}
 \end{align}
 The order parameters $C$, $R$, $G$, $D$, $m$, and $\hat m$ are ordinary numbers defined by
-\begin{align}
+\begin{align} \label{eq:ops.bos}
   C=\frac{\mathbf x\cdot\mathbf x}N
   &&
-  R=i\frac{\mathbf x\cdot\hat{\mathbf x}}N
+  R=-i\frac{\mathbf x\cdot\hat{\mathbf x}}N
   &&
   G=\frac{\bar{\pmb\eta}\cdot\pmb\eta}N
   &&
@@ -822,17 +822,17 @@ The order parameters $C$, $R$, $G$, $D$, $m$, and $\hat m$ are ordinary numbers
   &&
   m=\frac{\mathbf x_0\cdot\mathbf x}N
   &&
-  \hat m=i\frac{\mathbf x_0\cdot\hat{\mathbf x}}N
+  \hat m=-i\frac{\mathbf x_0\cdot\hat{\mathbf x}}N
 \end{align}
 while $\bar H$, $H$, $\bar{\hat H}$, $\hat H$, $\bar H_0$ and $H_0$ are Grassmann numbers defined by
-\begin{align}
+\begin{align} \label{eq:ops.ferm}
   \bar H=\frac{\bar{\pmb\eta}\cdot\mathbf x}N
   &&
   H=\frac{\pmb\eta\cdot\mathbf x}N
   &&
-  \bar{\hat H}=i\frac{\bar{\pmb\eta}\cdot\hat{\mathbf x}}N
+  \bar{\hat H}=-i\frac{\bar{\pmb\eta}\cdot\hat{\mathbf x}}N
   &&
-  \hat H=i\frac{\pmb\eta\cdot\hat{\mathbf x}}N
+  \hat H=-i\frac{\pmb\eta\cdot\hat{\mathbf x}}N
   &&
   \bar H_0=\frac{\bar{\pmb\eta}\cdot\mathbf x_0}N
   &&
@@ -1072,35 +1072,183 @@ 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 average number of stationary points}
+\section{Details for the average number of stationary points}
 \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_+\,d\mathbf a_-\,d\mathbf b_+\,d\mathbf b_-\, \notag \\
+  &\qquad\times\exp\Bigg\{
+    -\frac12\begin{bmatrix}\mathbf a_+^T&\mathbf b_+^T\end{bmatrix}\big(\partial\partial L(\mathbf x,\pmb\omega)+i\epsilon I\big)\begin{bmatrix}\mathbf a_+\\\mathbf b_+\end{bmatrix}
+    -\frac12\begin{bmatrix}\mathbf a_-^T&\mathbf b_-^T\end{bmatrix}\big(\partial\partial L(\mathbf x,\pmb\omega)-i\epsilon I\big)\begin{bmatrix}\mathbf a_-\\\mathbf b_-\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 $\mathbf a_\pm$ are in $\mathbb R^N$, the $\mathbf b_\pm$ are in
+$\mathbb R^M$, and the $\pmb\eta$ and $\pmb\gamma$ 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
-  +\frac i{\sqrt2}(\bar\theta_1\theta_1+\bar\theta_2\theta_2)\mathbf b
+  &\qquad+\frac1{\sqrt2}(\bar\theta_1\theta_2+\bar\theta_2\theta_1)\mathbf a_+
+  +\frac i{\sqrt2}(\bar\theta_1\theta_1+\bar\theta_2\theta_2)\mathbf a_-
   +\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)c_k
-  +\frac i{\sqrt2}(\bar\theta_1\theta_1+\bar\theta_2\theta_2)d_k
+  &\qquad+\frac1{\sqrt2}(\bar\theta_1\theta_2+\bar\theta_2\theta_1)b_{-k}
+  +\frac i{\sqrt2}(\bar\theta_1\theta_1+\bar\theta_2\theta_2)b_{+k}
   +\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\|^2-\|\mathbf b\|^2+\|\mathbf c\|^2-\|\mathbf d\|^2)
+   (\|\mathbf a_+\|^2-\|\mathbf a_-\|^2+\|\mathbf b_+\|^2-\|\mathbf b_-\|^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} and \eqref{eq:ops.ferm}, we also define
+\begin{align}
+  A_{++}=\frac{\mathbf a_+\cdot\mathbf a_+}N
+  &&
+  A_{--}=\frac{\mathbf a_-\cdot\mathbf a_-}N
+  &&
+  A_{+-}=\frac{\mathbf a_+\cdot\mathbf a_-}N
+  &&
+  X_+=\frac{\mathbf x\cdot\mathbf a_+}N
+  &&
+  X_-=\frac{\mathbf x\cdot\mathbf a_-}N
+  \\
+  \hat X_+=-i\frac{\hat{\mathbf x}\cdot\mathbf a_+}N
+  &&
+  \hat X_-=-i\frac{\hat{\mathbf x}\cdot\mathbf a_-}N
+  &&
+  H_{ij}=\frac{\bar{\pmb\eta}_i\cdot\pmb\eta_j}N
+  &&
+  J=\frac{\pmb\eta_1\cdot\pmb\eta_2}N
+  &&
+  \bar J=\frac{\bar{\pmb\eta}_1\cdot\bar{\pmb\eta}_2}N
+\end{align}
+These come with a volume term in the action
+\begin{equation}
+  \begin{aligned}
+    \frac12\log\det\left(
+      \begin{bmatrix}
+        C   & iR        & X_+       & X_-       \\
+        iR  & D         & i\hat X_+ & i\hat X_- \\
+        X_+ & i\hat X_+ & A_{++}    & A_{+-}    \\
+        X_- & i\hat X_- & A_{+-}    & A_{--}
+      \end{bmatrix}
+      -\begin{bmatrix}
+        m \\
+        i\hat m \\
+        m_+ \\
+        m_-
+      \end{bmatrix}
+      \begin{bmatrix}
+        m &
+        i\hat m &
+        m_+ &
+        m_-
+      \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_+=0 &&
+  X_-=0 &&
+  G_+=-R-\frac12(A_{++}+A_{--})
+\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}
+\
+
+
 
 \section{The quenched shattering energy in {\oldstylenums 1}\textsc{frsb} models}
 \label{sec:1frsb}
-- 
cgit v1.2.3-70-g09d2

