From cfa430efcf51b2f2032111d9eac1efd4520e59c5 Mon Sep 17 00:00:00 2001
From: Jaron Kent-Dobias <jaron@kent-dobias.com>
Date: Fri, 9 Jun 2023 17:49:00 +0200
Subject: Lots of writing and bibs.

---
 2-point.tex | 82 +++++++++++++++++++++++++++++++++++++++++--------------------
 1 file changed, 55 insertions(+), 27 deletions(-)

(limited to '2-point.tex')

diff --git a/2-point.tex b/2-point.tex
index bb7128a..00dc6ae 100644
--- a/2-point.tex
+++ b/2-point.tex
@@ -113,7 +113,7 @@ over a short parameter interval. These include actual (structural) glasses,
 spin glasses, certain inference and optimization problems, and more
 \cite{lots}. Glassiness is qualitatively understood to arise from structure of
 an energy or cost landscape, whether due to the proliferation of metastable
-states, or to the raising of barriers which cause effective dynamic constraints.
+states, or to the raising of barriers which cause effective dynamic constraints \cite{Cavagna_2001_Fragile}.
 However, in most models there is no quantitative correspondence between these
 landscape properties and the dynamic behavior they are purported to describe.
 
@@ -168,14 +168,12 @@ Franz--Parisi potential.
 \section{Model}
 \label{sec:model}
 
-\cite{Crisanti_1992_The, Crisanti_1993_The}
-
 The mixed spherical models are defined by the Hamiltonian
 \begin{equation} \label{eq:hamiltonian}
   H(\mathbf s)=-\sum_p\frac1{p!}\sum_{i_1\cdots i_p}^NJ^{(p)}_{i_1\cdots i_p}s_{i_1}\cdots s_{i_p}
 \end{equation}
 where the vectors $\mathbf s\in\mathbb R^N$ are confined to the sphere
-$\|\mathbf s\|^2=N$.  The coupling coefficients $J$ are fully-connected and random, with
+$\|\mathbf s\|^2=N$ \cite{Kirkpatrick_1987_p-spin-interaction, Crisanti_1992_The, Crisanti_2004_Spherical}.  The coupling coefficients $J$ are fully-connected and random, with
 zero mean and variance $\overline{(J^{(p)})^2}=a_pp!/2N^{p-1}$ scaled so that
 the energy is typically extensive. The overbar denotes an average
 over the coefficients $J$. The factors $a_p$ in the variances are freely chosen
@@ -195,7 +193,7 @@ where the function $f$ is defined from the coefficients $a_p$ by
 In this paper, we will focus on models with a replica symmetric complexity, but
 many of the intermediate formulae are valid for arbitrary replica symmetry
 breakings. At most {\oldstylenums1}\textsc{rsb} in the equilibrium is guaranteed if the function
-$\chi(q)=f''(q)^{-1/2}$ is convex. The complexity at the ground state must
+$\chi(q)=f''(q)^{-1/2}$ is convex \cite{Crisanti_1992_The}. The complexity at the ground state must
 reflect the structure of equilibrium, and therefore be replica symmetric. We
 are not aware of any result guaranteeing this for the complexity away from the
 ground state, but we check that our replica-symmetric solutions satisfy the
@@ -292,9 +290,11 @@ In this study, we focus exclusively on the model studied in
 \begin{equation}
   f_{3+4}(q)=\frac12\big(q^3+q^4\big)
 \end{equation}
-First, it has concave $f''(q)^{-1/2}$, so at least the ground state complexity
+First, it has convex $f''(q)^{-1/2}$, so at least the ground state complexity
 must be replica symmetric, and second, properties of its long-time dynamics
-have been extensively studied.
+have been extensively studied. The annealed one-point complexity of these
+models was calculated in \cite{BenArous_2019_Geometry}, and for this model the
+annealed is expected to be correct.
 
 \section{Results}
 \label{sec:results}
@@ -403,12 +403,12 @@ We must put a small caveat here: in \emph{any} situation, this calculation
 admits order-one other marginal minima to lie a subextensive distance from the
 reference point. For such a population of points, $\Sigma_{12}=0$ and $q=1$,
 which is always a permitted solution when at least one marginal direction
-exists. These points are of course separated by small barriers from one
-another, but they also cover a vanishing piece of configuration space, and each
-such cluster of points is isolated by extensive barriers from each other
-cluster in the way described above. To move on a `manifold' nearby marginal
-minima within such a cluster cannot describe aging, since the overlap with the
-initial condition will never change from one.
+exists. These points are separated by small barriers from one another, but they
+also cover a vanishing piece of configuration space, and each such cluster of
+points is isolated by extensive barriers from each other cluster in the way
+described above. To move on a `manifold' nearby marginal minima within such a
+cluster cannot describe aging, since the overlap with the initial condition
+will never change from one.
 
 \begin{figure}
   \includegraphics{figs/nearest_energies_above.pdf}
@@ -499,8 +499,11 @@ We introduce the Kac--Rice \cite{Kac_1943_On, Rice_1944_Mathematical} measure
   \delta\big(\nabla H(\mathbf s,\omega)\big)\,
   \big|\det\operatorname{Hess}H(\mathbf s,\omega)\big|
 \end{equation}
-which counts stationary points of the function $H$. More interesting is the
-measure conditioned on the energy density $E$ and stability $\mu$ of the points,
+which counts stationary points of the function $H$. If integrated over
+configuration space, $\mathcal N_H=\int d\nu_H(\mathbf s,\omega)$ gives the
+total number of stationary points in the function. The Kac--Rice method has been used by in many studies to analyze the geometry of random functions \cite{Cavagna_1998_Stationary, Fyodorov_2007_Density, Bray_2007_Statistics}. More interesting is the
+measure conditioned on the energy density $E$ and stability $\mu$ of the
+points,
 \begin{equation}
   d\nu_H(\mathbf s,\omega\mid E,\mu)
   =d\nu_H(\mathbf s,\omega)\,
@@ -515,7 +518,11 @@ Fig.~\ref{fig:spectra} for examples.
 
 We want the typical number of stationary points with energy density
 $E_1$ and stability $\mu_1$ that lie a fixed overlap $q$ from a reference
-stationary point of energy density $E_0$ and stability $\mu_0$.
+stationary point of energy density $E_0$ and stability $\mu_0$. For a
+\emph{typical} number, we cannot average the total number $\mathcal N_H$, which
+is exponentially large in $N$ and therefore can be biased by atypical examples.
+Therefore, we will average the log of this number. The two-point complexity is
+therefore defined by
 \begin{equation} \label{eq:complexity.definition}
   \Sigma_{12}
   =\frac1N\overline{\int\frac{d\nu_H(\pmb\sigma,\varsigma\mid E_0,\mu_0)}{\int d\nu_H(\pmb\sigma',\varsigma'\mid E_0,\mu_0)}\,
@@ -642,6 +649,8 @@ $\delta$ functions are promoted to exponentials can be evaluated by saddle
 point in the standard way, yielding an effective action depending on the above
 matrix fields alone.
 
+\subsection{Saddle point}
+
 We will always assume that the square matrices $C^{00}$, $R^{00}$, $D^{00}$,
 $C^{11}$, $R^{11}$, and $D^{11}$ are hierarchical matrices, with each set of
 three sharing the same hierarchical structure. In particular, we immediately
@@ -722,8 +731,13 @@ where we define for brevity (here and elsewhere) the constants
   v_f=f'(1)\big(f''(1)+f'''(1)\big)-f''(1)^2
 \end{align}
 Note that because the coefficients of $f$ must be nonnegative for $f$ to
-be a sensible covariance, both $u_f$ and $v_f$ are strictly positive. In
-general, $f^{(n)}(1)\geq f^{(m)}(1)$ if $n>m$.
+be a sensible covariance, both $u_f$ and $v_f$ are strictly positive. Note also
+that $u_f=v_f=0$ if $f$ is a homogeneous polynomial as in the pure models.
+These expressions are invalid for the pure models because $\mu_0$ and $E_0$
+cannot be fixed independently; we would have done the equivalent of inserting
+two identical $\delta$-functions! Instead, the terms $\hat\beta_0$ and
+$\hat\beta_1$ must be set to zero in our prior formulae (as if the energy was
+not constrained) and then the saddle point taken.
 
 
 In general, we except the $m\times n$ matrices $C^{01}$, $R^{01}$, $R^{10}$,
@@ -731,7 +745,7 @@ and $D^{01}$ to have constant \emph{rows} of length $n$, with blocks of rows
 corresponding to the \textsc{rsb} structure of the single-point complexity. For
 the scope of this paper, where we restrict ourselves to replica symmetric
 complexities, they have the following form at the saddle point:
-\begin{align}
+\begin{align} \label{eq:01.ansatz}
   C^{01}=
   \begin{subarray}{l}
   \hphantom{[}\begin{array}{ccc}\leftarrow&n&\rightarrow\end{array}\hphantom{\Bigg]}\\
@@ -1492,11 +1506,11 @@ properties as in Fig.~\ref{fig:min.neighborhood}.
 
 Here, we compute the Franz--Parisi potential for this model at zero
 temperature, with respect to a reference configuration fixed to be a stationary
-point of energy $E_0$ and stability $\mu_0$ as before.
-
-The Franz--Parisi 
-\cite{Franz_1995_Recipes, Franz_1998_EffectivePotential}
-
+point of energy $E_0$ and stability $\mu_0$ as before \cite{Franz_1995_Recipes,
+Franz_1998_EffectivePotential}. The potential is defined as the average free
+energy of a system constrained to lie with a fixed overlap $q$ with a reference
+configuration (here a stationary point with fixed energy and stability), and
+given by
 \begin{equation} \label{eq:franz-parisi.definition}
   \beta V_\beta(q\mid E_0,\mu_0)
   =-\frac1N\overline{\int\frac{d\nu_H(\pmb\sigma,\varsigma\mid E_0,\mu_0)}{\int d\nu_H(\pmb\sigma',\varsigma'\mid E_0,\mu_0)}\,
@@ -1507,6 +1521,10 @@ Both the denominator and the logarithm are treated using the replica trick, whic
   \beta V_\beta(q\mid E_0,\mu_0)
   =-\frac1N\lim_{\substack{m\to0\\n\to0}}\frac\partial{\partial n}\overline{\int\left(\prod_{b=1}^md\nu_H(\pmb\sigma_b,\varsigma_b\mid E_0,\mu_0)\right)\left(\prod_{a=1}^nd\mathbf s_a\,\delta(\|\mathbf s_a\|^2-N)\,\delta(\pmb \sigma_1\cdot \mathbf s_a-Nq)\,e^{-\beta H(\mathbf s_a)}\right)}
 \end{equation}
+The derivation of this proceeds in much the same way as for the complexity or
+the isolated eigenvalue. Once the $\delta$-functions are converted to
+exponentials, the $H$-dependant terms can be expressed by convolution with the
+linear operator
 \begin{equation}
   \mathcal O(\mathbf t)
   =\sum_a^m\delta(\mathbf t-\pmb\sigma_a)\left(
@@ -1515,10 +1533,13 @@ Both the denominator and the logarithm are treated using the replica trick, whic
     -\beta
     \sum_a^n\delta(\mathbf t-\mathbf s_a)
 \end{equation}
+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)}
 \end{equation}
-
+where $\mathcal S_0$ is the same as in \eqref{eq:one-point.action} and
 \begin{equation}
   n\mathcal S_{\mathrm{FP}}
   =\frac12\beta^2\sum_{ab}^nf(Q_{ab})
@@ -1530,7 +1551,11 @@ Both the denominator and the logarithm are treated using the replica trick, whic
     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,
+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}.
+Inserting these gives
 \begin{equation}
   \begin{aligned}
     \beta V_\beta&=\frac12\beta^2\big[f(1)-(1-x)f(q_1)-xf(q_0)\big]
@@ -1564,7 +1589,10 @@ The second case is when the inner statistical mechanics problem has replica symm
 \end{equation}
 Though the saddle point in $y_1$ can be evaluated in this expression, it
 delivers no insight. The final potential is found by taking the saddle over
-$z$, $y_1$, and $q_0$.
+$z$, $y_1$, and $q_0$. A plot comparing the result to the minimum energy
+saddles is found in Fig.~\ref{fig:franz-parisi}. As noted above, there is
+little qualitatively different from what was found in \cite{Ros_2019_Complexity}
+for the pure models.
 
 \section{Conclusion}
 \label{sec:conclusion}
-- 
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