Update on Overleaf.

1 files changed, 13 insertions, 9 deletions

diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index a22b7b2..54e2575 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -52,11 +52,15 @@ replica-symmetry breaking scheme that is well-defined, and corresponds directly
 to the topological characteristics of those minima.
 
 
-A more general question, of interest in optimization problems, is how to define a `threshold level'. This notion was introduced in Ref \cite{cugliandolo1993analytical} in the context of the $p$-spin model, as the energy at which the constant energy patches of phase-space percolate - hence 
-explaining why dynamics should relax to that level.
+Perhaps the most interesting application of this computation is in the context  of
+optimization problems, see for example \cite{gamarnik2021overlap,alaoui2022sampling,huang2021tight}. A question
+that appears there is how to define a `threshold level'. This notion was introduced  \cite{cugliandolo1993analytical} in the context of the $p$-spin model, as the energy at which the patches of the same  energy in phase-space percolate  - hence
+explaining why dynamics never go below that level.
 The notion of a `threshold' for more complex landscapes has later been 
-attempted several times, never to our knowledge in a clear and unambiguous 
-way. One of the purposes of this paper is to
+invoked several times, never to our knowledge in a clear and unambiguous 
+way. One of the purposes of this paper is to give a sufficiently detailed
+characterization of a general landscape so that a meaningful general notion
+of threshold may  be introduced - if this is at all possible.
 
 
 \section{The model}
@@ -967,9 +971,9 @@ P(q)=\frac1{\mathcal N^2}\sum_{\mathbf s_1\in\mathcal S}\sum_{\mathbf s_2\in\mat
 {\em This is the probability that two stationary points randomly drawn from the ensemble
 of stationary points happen to be at overlap $q$}
 
-It is
-straightforward to show that moments of this distribution are related to
-certain averages of the form. These are evaluated for a given energy, index, etc, but
+%It is straightforward to show that moments of this distribution are related to
+%certain averages of the form. 
+These are evaluated for a given energy, index, etc, but
 we shall omit these subindices for simplicity.
 
 \begin{equation}
@@ -1026,8 +1030,8 @@ Consider two independent pure $p$ spin models $H_{p_1}({\mathbf s})$ and $H_{p_2
 {\mathbf \sigma} \cdot {\mathbf s}$.
 The complexities are
 \begin{eqnarray}
-    e^{N\Sigma(e)}&=&\int de_1 de_2 \; e^{N[ \Sigma_1(e_1) + \Sigma_2(e_2) + O(\varepsilon) -\lambda N [(e_1+e_2)-e]}\nonumber \\
-    e^{-G(\hat \beta)}&=&\int de de_1 de_2 \; e^{N[-\hat \beta e+ \Sigma_1(e_1) + \Sigma_2(e_2) + O(\varepsilon) -\lambda N [(e_1+e_2)-e]}
+    e^{N\Sigma(e)}&=&\int de_1 de_2 d\lambda \; e^{N[ \Sigma_1(e_1) + \Sigma_2(e_2) + O(\varepsilon) -\lambda N [(e_1+e_2)-e]}\nonumber \\
+    e^{-G(\hat \beta)}&=&\int de de_1 de_2 d\lambda\; e^{N[-\hat \beta e+ \Sigma_1(e_1) + \Sigma_2(e_2) + O(\varepsilon) -\lambda N [(e_1+e_2)-e]}
 \end{eqnarray}
 The maximum is given by $\Sigma_1'=\Sigma_2'=\hat \beta$, provided it occurs in the phase
 in which both $\Sigma_1$ and $\Sigma_2$ are non-zero. The two systems are `thermalized',