Merge branch 'master' of https://git.overleaf.com/629a30c097d0b9f4b4f7a69d

2 files changed, 38 insertions, 9 deletions

diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib
index 2f7dd6d..d33028e 100644
--- a/frsb_kac-rice.bib
+++ b/frsb_kac-rice.bib
@@ -384,3 +384,28 @@
   publisher={APS}
 }
 
+
+@article{gamarnik2021overlap,
+  title={The overlap gap property and approximate message passing algorithms for $ p $-spin models},
+  author={Gamarnik, David and Jagannath, Aukosh},
+  journal={The Annals of Probability},
+  volume={49},
+  number={1},
+  pages={180--205},
+  year={2021},
+  publisher={Institute of Mathematical Statistics}
+}
+@article{huang2021tight,
+  title={Tight Lipschitz hardness for optimizing mean field spin glasses},
+  author={Huang, Brice and Sellke, Mark},
+  journal={arXiv preprint arXiv:2110.07847},
+  year={2021}
+}
+
+
+@article{alaoui2022sampling,
+  title={Sampling from the Sherrington-Kirkpatrick Gibbs measure via algorithmic stochastic localization},
+  author={Alaoui, Ahmed El and Montanari, Andrea and Sellke, Mark},
+  journal={arXiv preprint arXiv:2203.05093},
+  year={2022}
+}
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 09c3042..1427085 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}
@@ -968,9 +972,9 @@ 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}
@@ -1027,8 +1031,8 @@ 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',