From 4b9e19bbd81a390236b413a25f24770027d4d32c Mon Sep 17 00:00:00 2001
From: "kurchan.jorge" <kurchan.jorge@gmail.com>
Date: Wed, 13 Jul 2022 09:53:26 +0000
Subject: Update on Overleaf.

---
 frsb_kac-rice.bib | 26 ++++++++++++++++++++++++++
 frsb_kac-rice.tex |  6 +++---
 2 files changed, 29 insertions(+), 3 deletions(-)

diff --git a/frsb_kac-rice.bib b/frsb_kac-rice.bib
index fdcbdac..b4ae437 100644
--- a/frsb_kac-rice.bib
+++ b/frsb_kac-rice.bib
@@ -693,4 +693,30 @@ stochastic localization},
  doi = {10.1103/physrevlett.128.248001}
 }
 
+@article{kent2021complex,
+  title={Complex complex landscapes},
+  author={Kent-Dobias, Jaron and Kurchan, Jorge},
+  journal={Physical Review Research},
+  volume={3},
+  number={2},
+  pages={023064},
+  year={2021},
+  publisher={APS}
+}
+@article{kent2022analytic,
+  title={Analytic continuation over complex landscapes},
+  author={Kent-Dobias, Jaron and Kurchan, Jorge},
+  journal={arXiv preprint arXiv:2204.06072},
+  year={2022}
+}
+@article{cavagna1997structure,
+  title={Structure of metastable states in spin glasses by means of a three replica potential},
+  author={Cavagna, Andrea and Giardina, Irene and Parisi, Giorgio},
+  journal={Journal of Physics A: Mathematical and General},
+  volume={30},
+  number={13},
+  pages={4449},
+  year={1997},
+  publisher={IOP Publishing}
+}
 
diff --git a/frsb_kac-rice.tex b/frsb_kac-rice.tex
index 2d2e003..c31cb62 100644
--- a/frsb_kac-rice.tex
+++ b/frsb_kac-rice.tex
@@ -50,7 +50,7 @@ when the actual ground-state of the model was computed by Parisi with a
 different scheme, the Bray--Moore result was not exact, and the problem has
 been open ever since \cite{Parisi_1979_Infinite}.  To date, the program of
 computing the number of stationary points---minima, saddle points, and
-maxima---of a mean-field glass has been only carried out for a small subset of
+maxima---of  mean-field complex landscapes has been only carried out for a small subset of
 models, including most notably the (pure) $p$-spin model ($p>2$)
 \cite{Rieger_1992_The, Crisanti_1995_Thouless-Anderson-Palmer} and for similar
 energy functions inspired by molecular biology, evolution, and machine learning
@@ -1268,7 +1268,7 @@ exponentially many stationary points. A cartoon of this picture is shown in Fig.
   } \label{fig:cartoon}
 \end{figure}
 
-\subsubsection{A concrete example}
+\subsubsection{A tractable example}
 
 One can construct a schematic 2RSB model from two 1RSB models.
 Consider two independent pure models of size $N$ and with $p_1$-spin and $p_2$-spin couplings,
@@ -1411,7 +1411,7 @@ saddles of random mean-field landscapes, including systems with many steps of
 RSB.  For systems with full RSB, we find that minima are exponentially
 subdominant with respect to saddles at all energy densities above the ground
 state.  The solution contains valuable geometric information that has yet to be
-extracted in all detail.
+extracted in all detail, for example considering several copies of the system \cite{cavagna1997structure}, or the extension to complex variables  \cite{kent2021complex,kent2022analytic}. 
 
 A first and very important application of the method here is to perform the
 calculation for high dimensional spheres, where it would give us a clear
-- 
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