From dec15e5d61ed3a38ac64511b5324f7f4a4460246 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Thu, 27 Jun 2024 09:20:25 +0200
Subject: Work on appendix with ordinary complexity derivation.

---
 marginal.tex | 142 ++++++++++++++++++++++++++++++++++-------------------------
 1 file changed, 81 insertions(+), 61 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index f46ed38..038f214 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -1820,42 +1820,45 @@ full-RSB setting.
 
 Using the $\mathbb R^{N|2}$ superfields
 \begin{equation}
-  \pmb\phi_a(1)=\mathbf x+\bar\theta_1\pmb\eta+\bar{\pmb\eta}\theta_1+\bar\theta_1\theta_1\hat{\mathbf x},
+  \pmb\phi_a(1)=\mathbf x_a+\bar\theta_1\pmb\eta_a+\bar{\pmb\eta}_a\theta_1+\bar\theta_1\theta_1\hat{\mathbf x}_a,
 \end{equation}
 the replicated count of stationary points can be written
 \begin{equation}
   \begin{aligned}
     &\mathcal N(E,\mu)^n
-    =\int\prod_{a=1}^nd\hat\beta_a\,d\pmb\phi_a\,
-    \\
-    &\qquad\times\exp\left[
-      \hat\beta_a E-\frac12\int d1\,B_a(1)\sum_{k=1}^MV^k(\pmb\phi_a(1))^2
-    \right]
+    =\int d\hat\beta\prod_{a=1}^n\,d\pmb\phi_a\,
+    \exp\bigg[
+      N\hat\beta E \\
+    &\qquad-\frac12\int d1\,\left(
+        B(1)\sum_{k=1}^MV_k(\pmb\phi_a(1))^2
+        -\mu\big(\|\pmb\phi_a(1)\|^2-N\big)
+      \right)
+    \bigg]
   \end{aligned}
 \end{equation}
-for $B_a(1)=1-\hat\beta_a\bar\theta_1\theta_1$.
+for $B(1)=1-\hat\beta\bar\theta_1\theta_1$.
 The derivation of the complexity follows from here nearly identically to that
 in Appendix A.2 of \citeauthor{Fyodorov_2022_Optimization} with superoperations
 replacing standard ones \cite{Fyodorov_2022_Optimization}. First we insert
-Dirac $\delta$ functions to fix each of the $M$ energies $V^k(\pmb\phi_a(1))$ as
+Dirac $\delta$ functions to fix each of the $M$ energies $V_k(\pmb\phi_a(1))$ as
 \begin{equation} \label{eq:Vv.delta}
   \begin{aligned}
-    &\int dv^k_a\,\delta\big(V^k(\pmb\phi_a(1))-v^k_a(1)\big)
+    &\delta\big(V_k(\pmb\phi_a(1))-v_{ka}(1)\big)
     \\
-    &\quad=\int dv^k_a\,d\hat v^k_a\,\exp\left[i\int d1\,\hat v^k_a(1)\big(V^k(\pmb\phi_a(1))-v^k_a(1)\big)\right]
+    &=\int d\hat v_{ka}\,\exp\left[i\int d1\,\hat v_{ka}(1)\big(V_k(\pmb\phi_a(1))-v_{ka}(1)\big)\right]
   \end{aligned}
 \end{equation}
-The squared $V^k$ appearing in the energy can now be replaced by the variables
-$v^k$, leaving the only remaining dependence on the disordered $V$ in the
+The squared $V_k$ appearing in the energy can now be replaced by the variables
+$v_k$, leaving the only remaining dependence on the disordered $V$ in the
 contribution of \eqref{eq:Vv.delta}, which is linear. The average over the
 disorder can then be computed, which yields
 \begin{equation}
   \begin{aligned}
-    &\overline{\sum_{k=1}^M\sum_{a=1}^n\exp\left[i\int d1\,\hat v^k_a(1)V^k(\pmb\phi_a(1))\right]}
+    &\overline{\sum_{k=1}^M\sum_{a=1}^n\exp\left[i\int d1\,\hat v_{ka}(1)V_k(\pmb\phi_a(1))\right]}
     \\
     &
     =\exp\left[
-      -\frac12\sum_{k=1}^M\sum_{ab=1}^n\int d1\,d2\,\hat v_a^k(1)f\left(\frac{\pmb\phi_a(1)^T\pmb\phi_b(2)}N\right)\hat v_b^k(2)
+      -\frac12\sum_{k=1}^M\sum_{ab}^n\int d1\,d2\,\hat v_{ka}(1)f\left(\frac{\pmb\phi_a(1)\cdot\pmb\phi_b(2)}N\right)\hat v_{kb}(2)
     \right]
   \end{aligned}
 \end{equation}
@@ -1863,24 +1866,26 @@ The result is factorized in the indices $k$ and Gaussian in the superfields $v$
 and $\hat v$ with kernel
 \begin{equation}
   \begin{bmatrix}
-    B_a(1)\delta_{ab}\delta(1,2) & i\delta_{ab}\delta(1,2) \\
-    i\delta_{ab}\delta(1,2) & f\left(\frac{\pmb\phi_a(1)^T\pmb\phi_b(2)}N\right)
+    B(1)\delta_{ab}\delta(1,2) & i\delta_{ab}\delta(1,2) \\
+    i\delta_{ab}\delta(1,2) & f\left(\frac{\pmb\phi_a(1)\cdot\pmb\phi_b(2)}N\right)
   \end{bmatrix}
 \end{equation}
-Making the $M$ independent Gaussian integrals, we therefore have
+Making the $M$ independent Gaussian integrals, we find
 \begin{equation}
   \begin{aligned}
     &\mathcal N(E,\mu)^n
-    =\int\left(\prod_{a=1}^nd\hat\beta_a\,d\pmb\phi_a\right)
+    =\int d\hat\beta\left(\prod_{a=1}^nd\pmb\phi_a\right)
     \exp\bigg[
-      \sum_a^n\hat\beta_aE \\
-    &\qquad-\frac M2\log\operatorname{sdet}\left(
-        \delta_{ab}\delta(1,2)+B_a(1)f\left(\frac{\pmb\phi_a(1)^T\pmb\phi_b(2)}N\right)
+      nN\hat\beta E+\frac\mu2\sum_a^n\int d1\,\|\pmb\phi_a\|^2 \\
+    &\quad-\frac M2\log\operatorname{sdet}\left(
+        \delta_{ab}\delta(1,2)+B(1)f\left(\frac{\pmb\phi_a(1)\cdot\pmb\phi_b(2)}N\right)
       \right)
     \bigg]
   \end{aligned}
 \end{equation}
-We make a change of variables from the fields $\pmb\phi$ to matrices $\mathbb Q_{ab}(1,2)=\frac1N\pmb\phi_a(1)^T\pmb\phi_b(2)$. This transformation results in a change of measure of the form
+We make a change of variables from the fields $\pmb\phi$ to matrices $\mathbb
+Q_{ab}(1,2)=\frac1N\pmb\phi_a(1)\cdot\pmb\phi_b(2)$. This transformation results
+in a change of measure of the form
 \begin{equation}
   \prod_{a=1}^n d\pmb\phi_a=d\mathbb Q\,(\operatorname{sdet}\mathbb Q)^\frac N2
   =d\mathbb Q\,\exp\left[\frac N2\log\operatorname{sdet}\mathbb Q\right]
@@ -1889,18 +1894,20 @@ We therefore have
 \begin{equation}
   \begin{aligned}
     &\mathcal N(E,\mu)^n
-    =\int\left(\prod_{a=1}^nd\hat\beta_a\right)\,d\mathbb Q\,
-    \exp\bigg[
-      \sum_a^n\hat\beta_aE
-      +\frac N2\log\operatorname{sdet}\mathbb Q
+    =\int d\hat\beta\,d\mathbb Q\,
+    \exp\bigg\{
+      nN\hat\beta E+N\frac\mu2\operatorname{sTr}\mathbb Q
+    +\frac N2\log\operatorname{sdet}\mathbb Q
       \\
-    &\qquad-\frac M2\log\operatorname{sdet}\left(
-      \delta_{ab}\delta(1,2)+B_a(1)f(\mathbb Q_{ab}(1,2))
-      \right)
-    \bigg]
+    &\qquad
+    -\frac M2\log\operatorname{sdet}\left[
+      \delta_{ab}\delta(1,2)+B(1)f(\mathbb Q_{ab}(1,2))
+    \right]
+  \bigg\}
   \end{aligned}
 \end{equation}
-We now need to blow up our supermatrices into our physical order parameters. We have that
+We now need to blow up our supermatrices into our physical order parameters. We
+have from the definition of $\pmb\phi$ and $\mathbb Q$ that
 \begin{equation}
   \begin{aligned}
     &\mathbb Q_{ab}(1,2)
@@ -1913,7 +1920,7 @@ where $C$, $R$, $D$, and $G$ are the matrices defined in
 \eqref{eq:order.parameters}. Other possible combinations involving scalar
 products between fermionic and bosonic variables do not contribute at physical
 saddle points \cite{Kurchan_1992_Supersymmetry}. Inserting this expansion into
-the expression above and evaluating the superdeterminants, we find
+the expression above and evaluating the superdeterminants and supertrace, we find
 \begin{equation}
   \mathcal N(E,\mu)^n=\int d\hat\beta\,dC\,dR\,dD\,dG\,e^{nN\mathcal S_\mathrm{KR}(\hat\beta,C,R,D,G)}
 \end{equation}
@@ -1921,63 +1928,76 @@ where the effective action is given by
 \begin{widetext}
 \begin{equation}
   \begin{aligned}
-    &\mathcal S_\mathrm{KR}(\hat\beta,C,R,D,G)
-    =\hat\beta E-\frac1n\operatorname{Tr}(G+R)\mu
-    +\frac1n\frac12\Big(\log\det(CD+R^2)-\log\det G^2\Big)
+    \mathcal S_\mathrm{KR}(\hat\beta,C,R,D,G)
+    &=\hat\beta E+\lim_{n\to0}\frac1n\Bigg(-\mu\operatorname{Tr}(G+R)
+    +\frac12\log\det\big[G^{-2}(CD+R^2)\big]
+    +\alpha\log\det\big[I+G\odot f'(C)\big]
     \\
-    &-\frac1n\frac\alpha2\left\{\log\det\left[
+    &\qquad-\frac\alpha2\log\det\left[
       \Big(
-        f'(C)\odot D-\hat\beta I+(G\odot G-R\odot R)\odot f''(C)
+        f'(C)\odot D-\hat\beta I+(G^{\circ2}-R^{\circ2})\odot f''(C)
       \Big)f(C)
       +(I-R\odot f'(C))^2
-    \right]-\log\det(I+G\odot f'(C))^2\right\}
+    \right]\Bigg)
   \end{aligned}
 \end{equation}
-where $\odot$ gives the Hadamard or componentwise product between the matrices, while other products and powers are matrix products and powers.
+where $\odot$ gives the Hadamard or componentwise product between the matrices
+and $A^{\circ n}$ gives the Hadamard power of $A$, while other products and
+powers are matrix products and powers.
 
-In the case where $\mu$ is not specified, the model has a BRST symmetry whose
-Ward identities give $D=\hat\beta R$ and $G=-R$
-\cite{Annibale_2004_Coexistence, Kent-Dobias_2023_How}. Using these relations,
-the effective action becomes particularly simple:
+In the case where $\mu$ is not specified, we can make use of the BRST symmetry
+of Appendix~\ref{sec:brst} whose Ward identities give $D=\hat\beta R$ and
+$G=-R$. Using these relations, the effective action becomes particularly
+simple:
 \begin{equation}
-  \mathcal S(\hat\beta, C, R)
+  \mathcal S_\mathrm{KR}(\hat\beta, C, R)
   =
   \hat\beta E
-  +\lim_{n\to0}\frac1n\left[
-    -\frac\alpha2\log\det\left[
-      I-\hat\beta f(C)(I-R\odot f'(C))^{-1}
+  +\frac12\lim_{n\to0}\frac1n\Big(
+    \log\det(I+\hat\beta CR^{-1})
+    -\alpha\log\det\left[
+      I-\hat\beta f(C)\big(I-R\odot f'(C)\big)^{-1}
     \right]
-    +\frac12\log\det(I+\hat\beta CR^{-1})
-  \right]
+  \Big)
 \end{equation}
-This effective action is general for arbitrary matrices $C$ and $R$. When using
-a replica symmetric ansatz of $C_{ab}=\delta_{ab}+c_0(1-\delta_{ab})$ and
+This effective action is general for arbitrary matrices $C$ and $R$, and
+therefore arbitrary \textsc{rsb} order. When using a replica symmetric ansatz
+of $C_{ab}=\delta_{ab}+c_0(1-\delta_{ab})$ and
 $R_{ab}=r\delta_{ab}+r_0(1-\delta_{ab})$, the resulting function of
 $\hat\beta$, $c_0$, $r$, and $r_0$ is
 \begin{equation}
   \begin{aligned}
-    \mathcal S=
+    &\mathcal S_\mathrm{KR}(\hat\beta,c_0,r,r_0)=
     \hat\beta E
-    -\frac\alpha 2\left[
-      \log\left(1-\frac{\hat\beta\big(f(1)-f(c_0)\big)}{1-rf'(1)+r_0f'(c_0)}\right)
-      -\frac{\hat\beta f(c_0)+r_0f'(c_0)}{
-        1-\hat\beta\big(f(1)-f(c_0)\big)-rf'(1)+rf'(c_0)
-      }+\frac{r_0f'(c_0)}{1-rf'(1)+r_0f'(c_0)}
-    \right] \\
     +\frac12\left[
       \log\left(1+\frac{\hat\beta(1-c_0)}{r-r_0}\right)
       +\frac{\hat\beta c_0+r_0}{\hat\beta(1-c_0)+r-r_0}
       -\frac{r_0}{r-r_0}
     \right]
+    \\
+    &\qquad-\frac\alpha 2\left[
+      \log\left(1-\frac{\hat\beta\big(f(1)-f(c_0)\big)}{1-rf'(1)+r_0f'(c_0)}\right)
+      -\frac{\hat\beta f(c_0)+r_0f'(c_0)}{
+        1-\hat\beta\big(f(1)-f(c_0)\big)-rf'(1)+rf'(c_0)
+      }+\frac{r_0f'(c_0)}{1-rf'(1)+r_0f'(c_0)}
+    \right]
   \end{aligned}
 \end{equation}
+\end{widetext}
 When $f(0)=0$ as in the cases directly studied in this work, this further
-simplifies as $c_0=r_0=0$. Extremizing this expression with respect to the
+simplifies as $c_0=r_0=0$. The effective action is then
+\begin{equation}
+  \mathcal S_\mathrm{KR}(\hat\beta,r)=
+  \hat\beta E
+  +\frac12
+    \log\left(1+\frac{\hat\beta}{r}\right)
+  -\frac\alpha 2
+    \log\left(1-\frac{\hat\beta f(1)}{1-rf'(1)}\right)
+\end{equation}
+Extremizing this expression with respect to the
 order parameters $\hat\beta$ and $r$ produces the red line of dominant minima
 shown in Fig.~\ref{fig:ls.complexity}.
 
-\end{widetext}
-
 \bibliography{marginal}
 
 \end{document}
-- 
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