From f755a8a48df59edeb4cc2251b4a9923ad0b79046 Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Mon, 6 Jun 2022 10:15:20 +0000
Subject: Update on Overleaf.

---
 frsb_kac-rice.tex | 38 ++++++++++++++++----------------------
 1 file changed, 16 insertions(+), 22 deletions(-)

(limited to 'frsb_kac-rice.tex')

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index f3b0558..e02cab8 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -67,13 +67,11 @@ and therefore its spectrum is given by the Wigner semicircle with radius $\sqrt{
 \end{equation}
 and the spectrum of $\operatorname{Hess}H$ is this shifted by $\mu$, or $\rho(\lambda-\mu)$.
 
-The parameter $\mu$ fixes the spectrum of the hessian. By manipulating it, one
-can decide to find the complexity of saddles of a certain macroscopic index, or
-of minima with a certain harmonic stiffness. When $\mu$ is taken to be within
-the range $\pm\sqrt{4f''(1)}=\pm\mu_m$, the critical points are constrained to have
-index $\frac12N(1-\mu/\mu_m)$. When $\mu>\mu_m$, the critical
-points are minima whose sloppiest eigenvalue is $\mu-\mu_m$. Finally,
-when $\mu=\mu_m$, the critical points are marginal minima.
+The parameter $\mu$ fixes the spectrum of the hessian. When it is an integration variable,
+and one restricts the domain of all integrations to compute saddles of a certain macroscopic index, or
+of minima with a certain harmonic stiffness, its value is the `softest' mode that adapts to change the Hessian \cite{Fyodorov_2007_Replica}. When it is fixed, then the restriction of the index of saddles is `payed' by the realization of the eigenvalues of the Hessian, usually a
+`harder' mode.
+
 
 
 \subsection{What to expect?}
@@ -88,7 +86,7 @@ have, in the absence of coupling,  the same dependence, but are stretched to one
 \begin{equation}
     \Sigma_1(H_1)= \Sigma_o(H_1/\alpha_1) \qquad ; \qquad \Sigma_2(H_2)= \Sigma_o(H_2/\alpha_2) 
 \end{equation}
-Each system has a ground state energy $E_{gs}^{1,2}$, a threshold energy $E_{thres}^{1,2}$ (a well-defined notion, since we are considering pure p-spins), abd the corresponding limit values $X^{1,2}_{gs}=\left. \frac{d \Sigma_1}{dE_{1,2}}\right|_{E^{gs}_{12}}$
+Each system has a ground state energy $E_{gs}^{1,2}$, a threshold energy $E_{thres}^{1,2}$ (a well-defined notion, since we are considering pure p-spins),  the corresponding limit values $X^{1,2}_{gs}=\left. \frac{d \Sigma_1}{dE_{1,2}}\right|_{E^{gs}_{12}}$
 and $X^{1,2}_{thres}=\left. \frac{d \Sigma_1}{dE_{1,2}}\right|_{E^{thres}_{12}}$
 Considering the cartesian product of both systems, we have, in terms of the total energy
 $H=H_1+H_2$ three regimes:
@@ -98,12 +96,10 @@ $H=H_1+H_2$ three regimes:
 & & X_1 \equiv \frac{d \Sigma_1}{dE_1}=  X_2 \equiv \frac{d \Sigma_2}{dE_2}
  \end{eqnarray}
 \item  {\bf Semi-frozen}
-As we go down in energy, one of the systems (say, the first) reaches its ground state,
-At lower temperatures, the first system is thus frozen, while the second is not,
-so that $X_1=X_1^{gs}> X_2$. The lowest energy is such that both systems are frozen.
+As we go down in energy, one of the systems (say, the first) reaches its frozen phase,
+ the first system is thus concentrated in a few states of $O(1)$ energy, while the second is not, so that $X_1=X_1^{gs}> X_2$. The lowest energy is reached when  systems are frozen.
 \item {\bf Semi-threshold }  As we go up from the unfrozen upwards in energy,
-the second system reaches its threshold $X_2^{thres}$. At higher energies minima are extremely rare,
-so the second system remains stuck at its threshold for higher energies.
+the second system reaches its threshold $X_2^{thres}$. At higher energies minima are extremely rare, so the minima of the second system remain stuck at its threshold for higher energies.
 
 \item{\bf Both systems reach their thresholds} There essentially no more minima above that.
 \end{itemize}
@@ -111,13 +107,12 @@ Consider now two combined vectors $({\bf s},{\bf \hat s})$ and $({\bf s}',{\bf \
 chosen at the same energies.\\
 
 $\bullet$ Their normalized overlap is close to one when both subsystems are frozen,
-close to a half in the semifrozen phase, and zero at all higher energies.\\
+between zero and one  in the semifrozen phase, and zero at all higher energies.\\
 
 $\bullet$ In phases where one or both systems are stuck in their thresholds (and only in those), the
-minima are exponentially subdominant with respect to saddles.
+minima are exponentially subdominant with respect to saddles, because a saddle is found by releasing the constraint of staying on the threshold.
+
 
-$\bullet$ {\bf note that the same reasoning leads us to the conclusion that
-minima of two total energies such that one of the systems is frozen have nonzero overlaps}
 
 \section{Main result}
 
@@ -285,7 +280,7 @@ $F$ is a $k-1$ RSB ansatz with all eigenvalues scaled by $y$ and shifted by $z$.
 
 
 
-\section{Kac--Rice}
+\section{Kac-Rice}
 
 \cite{Auffinger_2012_Random, BenArous_2019_Geometry}
 
@@ -475,10 +470,9 @@ The odd and even fermion numbers decouple, so we can neglect all odd terms in $\
 
 The variables $\bar \theta \theta$ and $\bar \theta ' \theta'$ play
 the role of `times' in a superspace treatment. We have a long experience of
-making an ansatz for replicated quantum problems, which naturally involve a (Matsubara) time. The analogy strongly
-suggests that only the diagonal ${\bf Q}_{aa}$ depend on the $\theta$'s. This boils down the ansatz \ref{ansatz}
-
-Not surprisingly, this ansatz closes, as we shall see. That it closes under Hadamard products is simple.
+making an ansatz for replicated quantum problems, which naturally involve a (Matsubara) time. The dependence on this time only holds for diagonal replica elements, a consequence of ultrametricity.  The analogy strongly
+suggests that only the diagonal ${\bf Q}_{aa}$ depend on the $\theta$'s. This boils down the ansatz \ref{ansatz}.
+Not surprisingly, and for the same reason as in the quantum case, this ansatz closes, as we shall see.For example, consider the convolution:
 
 \begin{equation}
   \begin{aligned}
-- 
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