More tweaks, citations.

2 files changed, 66 insertions, 10 deletions

diff --git a/topology.bib b/topology.bib
index 3281fd9..f28564a 100644
--- a/topology.bib
+++ b/topology.bib
@@ -151,6 +151,62 @@
  eprinttype = {arxiv}
 }
 
+@article{Castellani_2005_Spin-glass,
+ author = {Castellani, Tommaso and Cavagna, Andrea},
+ title = {Spin-glass theory for pedestrians},
+ journal = {Journal of Statistical Mechanics: Theory and Experiment},
+ publisher = {IOP Publishing},
+ year = {2005},
+ month = {5},
+ number = {05},
+ volume = {2005},
+ pages = {P05012},
+ url = {https://doi.org/10.1088%2F1742-5468%2F2005%2F05%2Fp05012},
+ doi = {10.1088/1742-5468/2005/05/p05012}
+}
+
+@article{Crisanti_1992_The,
+ author = {Crisanti, A. and Sommers, H.-J.},
+ title = {The spherical $p$-spin interaction spin glass model: the statics},
+ journal = {Zeitschrift für Physik B Condensed Matter},
+ publisher = {Springer Science and Business Media LLC},
+ year = {1992},
+ month = {10},
+ number = {3},
+ volume = {87},
+ pages = {341--354},
+ url = {https://doi.org/10.1007%2Fbf01309287},
+ doi = {10.1007/bf01309287}
+}
+
+@article{Crisanti_2004_Spherical,
+ author = {Crisanti, A. and Leuzzi, L.},
+ title = {Spherical $2+p$ Spin-Glass Model: An Exactly Solvable Model for Glass to Spin-Glass Transition},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {2004},
+ month = {11},
+ number = {21},
+ volume = {93},
+ pages = {217203},
+ url = {https://doi.org/10.1103%2Fphysrevlett.93.217203},
+ doi = {10.1103/physrevlett.93.217203}
+}
+
+@article{Crisanti_2006_Spherical,
+ author = {Crisanti, A. and Leuzzi, L.},
+ title = {Spherical $2+p$ spin-glass model: An analytically solvable model with a glass-to-glass transition},
+ journal = {Physical Review B},
+ publisher = {American Physical Society (APS)},
+ year = {2006},
+ month = {1},
+ number = {1},
+ volume = {73},
+ pages = {014412},
+ url = {https://doi.org/10.1103%2Fphysrevb.73.014412},
+ doi = {10.1103/physrevb.73.014412}
+}
+
 @book{DeWitt_1992_Supermanifolds,
  author = {DeWitt, Bryce S.},
  title = {Supermanifolds},
diff --git a/topology.tex b/topology.tex
index baf0766..212a20d 100644
--- a/topology.tex
+++ b/topology.tex
@@ -56,9 +56,9 @@ $\star$ \href{mailto:jaron.kent-dobias@roma1.infn.it}{\small jaron.kent-dobias@r
 
 \section*{\color{scipostdeepblue}{Abstract}}
 \textbf{\boldmath{%
-We consider the set of solutions to $M$ random polynomial equations of $N$
-variables. Each equation has independent Gaussian coefficients and a target
-value $V_0$, and their variables are restricted to the $(N-1)$-sphere. When
+We consider the set of solutions to $M$ random polynomial equations whose $N$
+variables are restricted to the $(N-1)$-sphere. Each equation has independent Gaussian coefficients and a target
+value $V_0$. When
 solutions exist, they form a manifold. We compute the average Euler
 characteristic of this manifold in the limit of large $N$, and find different
 behavior depending on the target value $V_0$, the ratio $\alpha=M/N$, and the
@@ -69,9 +69,9 @@ and there is a single extensive connected component, while in the onset phase
 the Euler characteristic is exponentially large in $N$. In the shattered phase
 the characteristic remains exponentially large but subextensive components
 appear, while in the \textsc{unsat} phase the manifold vanishes. When $M=1$
-there is a correspondence between this problem and the topology of
-energy level sets in the spherical spin glasses. We conjecture that
-the transition from the onset to shattered phase corresponds to the asymptotic
+there is a correspondence between this problem and
+level sets of the energy in the spherical spin glasses. We conjecture that
+the transition between the onset and shattered phases corresponds to the asymptotic
 limit of gradient descent from a random initial condition.
 }
 }
@@ -165,7 +165,7 @@ level set of a spherical spin glass with energy density $E=V_0/\sqrt{N}$.
 
 
 This problem or small variations thereof have attracted attention recently for
-their resemblance to encryption, optimization, and vertex models of confluent
+their resemblance to encryption, least-squares optimization, and vertex models of confluent
 tissues \cite{Fyodorov_2019_A, Fyodorov_2020_Counting,
   Fyodorov_2022_Optimization, Tublin_2022_A, Vivo_2024_Random, Urbani_2023_A, Kamali_2023_Dynamical,
   Kamali_2023_Stochastic, Urbani_2024_Statistical, Montanari_2023_Solving,
@@ -712,7 +712,7 @@ for the energies at which level sets of the spherical spin glasses have
 disconnected pieces appear, and that at which a large connected component
 vanishes. For the pure $p$-spin spherical spin glasses with $f(q)=\frac12q^p$,
 $E_\text{sh}=\sqrt{2(p-1)/p}$, precisely the threshold energy in these
-models. This is expected, since threshold energy, defined as the place where
+models \cite{Castellani_2005_Spin-glass}. This is expected, since threshold energy, defined as the place where
 marginal minima are dominant in the landscape, is widely understood as the
 place where level sets are broken into pieces.
 
@@ -1215,7 +1215,7 @@ orders in $N$, since for odd-dimensional manifolds the Euler characteristic is
 always zero. When $N+M+1$ is even, we have $\overline{\chi(\Omega)}=2$ to
 leading order in $N$, as specified in the main text.
 
-\section{Details for the average number of stationary points}
+\section{Details of the calculation of the complexity}
 \label{sec:complexity.details}
 
 Starting from \eqref{eq:abs.kac-rice}, we make the substitution
@@ -1554,7 +1554,7 @@ for arbitrary \textsc{rsb} structure in the matrix $C$ as
   \right]
 \end{equation}
 where $\chi(q)=\int_1^qdq'\int_0^{q'}dq''P(q'')$ and $P(q)$ is the distribution of
-off-diagonal elements of the matrix $C$. This action must be extremized over
+off-diagonal elements of the matrix $C$ \cite{Crisanti_1992_The, Crisanti_2004_Spherical, Crisanti_2006_Spherical}. This action must be extremized over
 the function $\chi$ and the variables $\hat\beta$ and $\tilde r_d$, under the
 constraint that $\chi(q)$ is continuous, that it has $\chi'(1)=-1$, and
 $\chi(1)=0$, necessary for $P$ to be a well-defined probability distribution.