From 7546d6e95ddaff512fbc538cf2cd2416d500c34b Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 7 Jun 2024 16:45:03 +0200
Subject: Lots of writing in the appendix.

---
 marginal.tex | 236 ++++++++++++++++++++++++++++++++++++++++++++---------------
 1 file changed, 178 insertions(+), 58 deletions(-)

(limited to 'marginal.tex')

diff --git a/marginal.tex b/marginal.tex
index 7df99d5..c461f81 100644
--- a/marginal.tex
+++ b/marginal.tex
@@ -238,8 +238,10 @@ with the effective action
 We need to evaluate the integral above using the saddle point method, but in the limit of $\beta\to\infty$.
 We expect the overlaps to concentrate on one as $\beta$ goes to infinity. We therefore take
 \begin{align}
+  \label{eq:q0.limit}
   q_0&=1-y\beta^{-1}-z\beta^{-2}+O(\beta^{-3})
   \\
+  \label{eq:q0t.limit}
   \tilde q_0&=1-\tilde y\beta^{-1}-(z+\Delta z)\beta^{-2}+O(\beta^{-3})
 \end{align}
 However, taking the limit with $y\neq\tilde y$ results in an expression for the
@@ -506,7 +508,8 @@ $\mathcal O$ then the average gives
 \end{equation}
 The result is an integral that only depends on the many vector variables we
 have introduced through their scalar products with each other. We therefore make a change of variables in the integration from those vectors to matrices that encode their possible scalar products. These matrices are
-\begin{align}
+\begin{equation} \label{eq:order.parameters}
+  \begin{aligned}
   C_{ab}=\frac1N\mathbf x_a\cdot\mathbf x_b
   &&
   R_{ab}=-i\frac1N\mathbf x_a\cdot\hat{\mathbf x}_b
@@ -520,7 +523,8 @@ have introduced through their scalar products with each other. We therefore make
   X^c_{ab}=\frac1N\mathbf x_a\cdot\mathbf s_b^c
   \\
   \hat X^c_{ab}=\frac1N\hat{\mathbf x}_a\cdot\mathbf s_b^c
-\end{align}
+  \end{aligned}
+\end{equation}
 Order parameters that mix the normal and Grassmann variables generically vanish
 in these settings \cite{Kurchan_1992_Supersymmetry}.
 
@@ -843,7 +847,7 @@ When expanded, this supermatrix is constructed of the scalar products of the
 real and Grassmann vectors that make up $\pmb\phi$. The change of variables to
 these order parameters again results in the Jacobian of \eqref{eq:coordinate.jacobian}, contributing
 \begin{equation}
-  \frac N2\log\det J(C,R,D,G,Q,X,\hat X)
+  \frac N2\log\det J(C,R,D,Q,X,\hat X)-\frac N2\log\det G^2
 \end{equation}
 Up to this point, the expressions above are general and independent of a given
 ansatz. However, we expect that the order parameters $X$ and $\hat X$ are zero,
@@ -881,56 +885,52 @@ Now we have reduced the problem to an extremal one over the order parameters
 $\hat\beta$, $\hat\lambda$, $C$, $R$, $D$, $G$, and $Q$, it is time to make an
 ansatz for the form of order we expect to find. We will focus on a regime where
 the structure of stationary points is replica symmetric, and further where
-typical pairs of stationary points have no overlap. This gives
+typical pairs of stationary points have no overlap. This requires that $f(0)=0$, or that there is no constant term in the random polynomials. This gives
 \begin{align}
-  C=I && R=r_dI && D = d_dI && G = g_dI
+  C=I && R=rI && D = dI && G = gI
 \end{align}
 We further take a planted replica symmetric structure for the matrix $Q$,
-identical to that in \eqref{eq:Q.structure}.
+identical to that in \eqref{eq:Q.structure}. The resulting effective action is
+the same as if we had made an annealed calculation in the complexity, though
+the previous expressions are general.
 \begin{equation}
   \begin{aligned}
-    \mathcal S
-    =-\frac\alpha2\log\left[
+    \mathcal S_\beta
+    =\hat\beta E-\mu(r+g)
+    +\frac12\log\frac{d+r^2}{g^2}\frac{1-2q_0+\tilde q_0^2}{(1-q_0)^2}
+    -\frac\alpha2\log\left(\frac{1-f'(1)(2\beta(1-q_0)+\hat\lambda-(1-2q_0+\tilde q_0^2)\beta(\beta+\hat\lambda)f'(1))}{(1-(1-q_0)\beta f'(1))^2}\right)
+    \\
+    -\frac12\mu\hat\lambda+\hat\lambda\lambda^*-\frac\alpha2\log\left[
       \frac{
-        (f'(1)d-\hat\beta-f''(1)(r^2-g^2+q_0^2\beta^2-\tilde q_0^2\beta(\beta+\hat\lambda)+\beta\hat\lambda+\frac12\hat\lambda^2)))(f(1)-f(0))+(1-rf'(1))^2)
+        \big[f'(1)d-\hat\beta-f''(1)(r^2-g^2+q_0^2\beta^2-\tilde q_0^2\beta(\beta+\hat\lambda)+\beta\hat\lambda+\frac12\hat\lambda^2)\big]f(1)+(1-rf'(1))^2
       }{
         (1+gf'(1))^2
       }
-    \right] \\
-    +\frac{\alpha f(0)}2\frac{
-      \hat\beta-df'(1)+(r^2-g^2+q_0^2\beta^2-\tilde q_0^2\beta(\beta+\lambda)+\lambda\beta+\frac12\lambda^2)f''(1)
-    }{
-      (f'(1)d-\hat\beta- (r^2-g^2+q_0^2\beta^2-\tilde q_0^2\beta(\beta+\hat\lambda)+\beta\hat\lambda+\frac12\hat\lambda^2)f''(1))(f(1)-f(0))+(1-rf'(1))^2
-    }
-    \\
-    -\frac\alpha2\log\left(\frac{1-f'(1)(2\beta(1-q_0)+\lambda-(1-2q_0+\tilde q_0^2)\beta(\beta+\lambda)f'(1))}{(1-(1-q_0)\beta f'(1))^2}\right)
-    +\frac12\log\frac{d+r^2}{g^2}\frac{1-2q_0+\tilde q_0^2}{(1-q_0)^2}
-    -\mu(r+g)-\frac12\mu\hat\lambda+\hat\beta E+\hat\lambda\lambda^*
+    \right]
   \end{aligned}
 \end{equation}
+We expect as before the limits of $q_0$ and $\tilde q_0$ as $\beta$ goes to
+infinity to approach one, defining their asymptotic expansion as in
+\eqref{eq:q0.limit} and \eqref{eq:q0t.limit}. Upon making this substitution and
+taking the zero-temperature limit, we find
 \begin{equation}
   \begin{aligned}
     \mathcal S_\infty
-    =-\frac\alpha2\log\left[
-      \frac{
-        (f'(1)d-\hat\beta-f''(1)(r^2-g^2+2y_0\hat\lambda+\Delta z+\frac12\hat\lambda^2)))(f(1)-f(0))+(1-rf'(1))^2)
-      }{
-        (1+gf'(1))^2
-      }
-    \right] \\
-    +\frac{\alpha f(0)}2\frac{
-      \hat\beta-df'(1)+(r^2-g^2+2y_0\hat\lambda+\Delta z+\frac12\lambda^2)f''(1)
-    }{
-      (f'(1)d-\hat\beta- (r^2-g^2+2y_0\hat\lambda+\Delta z+\frac12\lambda^2)f''(1))(f(1)-f(0))+(1-rf'(1))^2
-    }
-    \\
+    =\hat\beta E-\mu(r+g)
+    +\frac12\log\frac{d+r^2}{g^2}\frac{y_0^2-\Delta z}{y_0^2}
     -\frac\alpha2\log\left(
       \frac{
         1-(2y_0+\hat\lambda)f'(1)+(y_0^2-\Delta z)f'(1)^2
       }{(1-y_0f'(1))^2}
     \right)
-    +\frac12\log\frac{d+r^2}{g^2}\frac{y_0^2-\Delta z}{y_0^2}
-    -\mu(r+g)-\frac12\mu\hat\lambda+\hat\beta E+\hat\lambda\lambda^*
+    \\
+    -\frac12\mu\hat\lambda+\hat\lambda\lambda^*-\frac\alpha2\log\left[
+      \frac{
+        \big[f'(1)d-\hat\beta-f''(1)(r^2-g^2+2y_0\hat\lambda+\Delta z+\frac12\hat\lambda^2)\big]f(1)+\big[1-rf'(1)\big]^2
+      }{
+        (1+gf'(1))^2
+      }
+    \right]
   \end{aligned}
 \end{equation}
 \begin{equation}
@@ -950,43 +950,163 @@ replica symmetric structure, formulas for the effective action are generic to
 any structure and provide a starting point for analyzing the challenging
 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},
+\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]
+  \end{aligned}
+\end{equation}
+for $B_a(1)=1-\hat\beta_a\bar\theta_1\theta_1$.
+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
+\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)
+    \\
+    &\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]
+  \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
+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]}
+    \\
+    &
+    =\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)
+    \right]
+  \end{aligned}
+\end{equation}
+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)
+  \end{bmatrix}
+\end{equation}
+where $\delta(1,2)=(\bar\theta_1-\bar\theta_2)(\theta_1-\theta_2)$ is the
+identity operator for convolutions with $d1$ or $d2$.
+Making the $M$ independent Gaussian integrals, we therefore have
+\begin{equation}
+  \begin{aligned}
+    &\mathcal N(E,\mu)^n
+    =\int\left(\prod_{a=1}^nd\hat\beta_a\,d\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)
+      \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
+\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]
+\end{equation}
+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
+      \\
+    &\qquad-\frac M2\log\operatorname{sdet}\left(
+      \delta_{ab}\delta(1,2)+B_a(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
+\begin{equation}
+  \begin{aligned}
+    &\mathbb Q_{ab}(1,2)
+    =C_{ab}-G_{ab}(\bar\theta_1\theta_2+\bar\theta_2\theta_1) \\
+    &\qquad-R_{ab}(\bar\theta_1\theta_1+\bar\theta_2\theta_2)
+    -D_{ab}\bar\theta_1\theta_2\bar\theta_2\theta_2
+  \end{aligned}
+\end{equation}
+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
+\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}
+where the effective action is given by
 \begin{widetext}
 \begin{equation}
   \begin{aligned}
-    &\mathcal S
-    =-\frac1n\frac\alpha2\left\{\log\det\left[
-      \hat\beta f(C)+\Big(
-        f'(C)\odot D+(G\odot G-R\odot R)\odot f''(C)
+    &\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)
+    \\
+    &-\frac1n\frac\alpha2\left\{\log\det\left[
+      \Big(
+        f'(C)\odot D-\hat\beta I+(G\odot G-R\odot R)\odot f''(C)
       \Big)f(C)
-      +(I+R\odot f'(C))^2
-    \right]-\log\det(I+G\odot f'(C))^2\right\} \\
-    &+\frac1n\frac12\Big(\log\det(CD+R^2)-\log\det G^2\Big)
-    +\hat\beta E+(g_d-r_d)\mu
+      +(I-R\odot f'(C))^2
+    \right]-\log\det(I+G\odot f'(C))^2\right\}
   \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.
 
+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:
+\begin{equation}
+  \mathcal S(\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}
+    \right]
+    +\frac12\log\det(I+\hat\beta CR^{-1})
+  \right]
+\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
+$R_{ab}=r\delta_{ab}+r_0(1-\delta_{ab})$, the resulting function of
+$\hat\beta$, $c_0$, $r$, and $r_0$ is
 \begin{equation}
   \begin{aligned}
-    &\hat\beta E+\mu(g_d-r_d)+\frac12\log\frac{d_d+r_d^2}{g_d^2} \\
-    &-\frac\alpha2\log\left[
-      1+\hat\beta\big(f(1)-f(0)\big)
-      \Big(d_d\big(f(1)-f(0)\big)+r_d\big(2+r_df'(1)\big)\Big)f'(1)
-      +(g_d^2-r_d^2)\big(f(1)-f(0)\big)f''(1)
+    \mathcal S=
+    \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] \\
-    &-\alpha f(0)\left(
-      \big(f(1)-f(0)\big)+\frac{1+r_d\big(2+r_df'(1)\big)f'(1)}{\hat\beta+d_df'(1)+(g_d^2-r_d^2)f''(1)}
-    \right)^{-1}
+    +\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]
   \end{aligned}
 \end{equation}
-
-In the case where $\mu$ is not specified, in which the model is supersymmetric, $D=\hat\beta R$ and the effective action becomes particularly simple:
-\begin{equation}
-  \hat\beta e
-  -\frac12\frac{\alpha f(0)}{1+\hat\beta\big(f(1)-f(0)\big)+r_df'(1)}
-  -\frac\alpha2\log\left(1+\frac{\hat\beta\big(f(1)-f(0)\big)}{1+r_df'(1)}\right)
-  +\frac12\log\frac{\hat\beta+r_d}{r_d}
-\end{equation}
+When $f(0)=0$ as in the cases directly studied in this work, this further
+simplifies as $c_0=r_0=0$.
 
 \end{widetext}
 
-- 
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