Merged old article into new without much editing.

14 files changed, 696 insertions, 5 deletions

diff --git a/.gitignore b/.gitignore
index bc6c252..eebab43 100644
--- a/.gitignore
+++ b/.gitignore
@@ -1,4 +1,5 @@
 *.aux
+*.toc
 *.bbl
 *.blg
 *.fdb_latexmk
diff --git a/figs/complexity.pdf b/figs/complexity.pdf
new file mode 100644
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diff --git a/figs/threshold_2.000.pdf b/figs/threshold_2.000.pdf
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diff --git a/stokes.bib b/stokes.bib
index 497686c..50aa4c7 100644
--- a/stokes.bib
+++ b/stokes.bib
@@ -1,3 +1,150 @@
+@article{Anninos_2016_Disordered,
+ author = {Anninos, Dionysios and Anous, Tarek and Denef, Frederik},
+ title = {Disordered quivers and cold horizons},
+ journal = {Journal of High Energy Physics},
+ publisher = {Springer Science and Business Media LLC},
+ year = {2016},
+ month = {12},
+ volume = {2016},
+ pages = {71},
+ url = {https://doi.org/10.1007%2Fjhep12%282016%29071},
+ doi = {10.1007/jhep12(2016)071}
+}
+
+@article{Antenucci_2015_Complex,
+ author = {Antenucci, F. and Crisanti, A. and Leuzzi, L.},
+ title = {Complex spherical {$2+4$} spin glass: A model for nonlinear optics in random media},
+ journal = {Physical Review A},
+ publisher = {American Physical Society (APS)},
+ year = {2015},
+ month = {5},
+ number = {5},
+ volume = {91},
+ pages = {053816},
+ url = {https://doi.org/10.1103%2Fphysreva.91.053816},
+ doi = {10.1103/physreva.91.053816}
+}
+
+@article{Auffinger_2012_Random,
+ author = {Auffinger, Antonio and Ben Arous, Gérard and Černý, Jiří},
+ title = {Random Matrices and Complexity of Spin Glasses},
+ journal = {Communications on Pure and Applied Mathematics},
+ publisher = {Wiley},
+ year = {2012},
+ month = {9},
+ number = {2},
+ volume = {66},
+ pages = {165--201},
+ url = {https://doi.org/10.1002%2Fcpa.21422},
+ doi = {10.1002/cpa.21422}
+}
+
+@article{Auffinger_2013_Complexity,
+ author = {Auffinger, Antonio and Ben Arous, Gérard},
+ title = {Complexity of random smooth functions on the high-dimensional sphere},
+ journal = {The Annals of Probability},
+ publisher = {Institute of Mathematical Statistics},
+ year = {2013},
+ month = {11},
+ number = {6},
+ volume = {41},
+ pages = {4214--4247},
+ url = {https://doi.org/10.1214%2F13-aop862},
+ doi = {10.1214/13-aop862}
+}
+
+@article{Behtash_2017_Toward,
+ author = {Behtash, Alireza and Dunne, Gerald V. and Schäfer, Thomas and Sulejmanpasic, Tin and Ünsal, Mithat},
+ title = {Toward {Picard}–{Lefschetz} theory of path integrals, complex saddles and resurgence},
+ journal = {Annals of Mathematical Sciences and Applications},
+ publisher = {International Press of Boston},
+ year = {2017},
+ number = {1},
+ volume = {2},
+ pages = {95--212},
+ url = {https://doi.org/10.4310%2Famsa.2017.v2.n1.a3},
+ doi = {10.4310/amsa.2017.v2.n1.a3}
+}
+
+@book{Bezout_1779_Theorie,
+ author = {Bézout, Etienne},
+ title = {Théorie générale des équations algébriques},
+ publisher = {L'imprimerie de Ph.-D. Pierres},
+ year = {1779},
+ url = {https://archive.org/details/thoriegnra00bz/},
+ address = {rue S. Jacques, Paris}
+}
+
+@article{Bogomolny_1992_Distribution,
+ author = {Bogomolny, E. and Bohigas, O. and Leboeuf, P.},
+ title = {Distribution of roots of random polynomials},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {1992},
+ month = {5},
+ number = {18},
+ volume = {68},
+ pages = {2726--2729},
+ url = {https://doi.org/10.1103%2Fphysrevlett.68.2726},
+ doi = {10.1103/physrevlett.68.2726}
+}
+
+@article{Bray_1980_Metastable,
+ author = {Bray, A J and Moore, M A},
+ title = {Metastable states in spin glasses},
+ journal = {Journal of Physics C: Solid State Physics},
+ publisher = {IOP Publishing},
+ year = {1980},
+ month = {7},
+ number = {19},
+ volume = {13},
+ pages = {L469--L476},
+ url = {https://doi.org/10.1088%2F0022-3719%2F13%2F19%2F002},
+ doi = {10.1088/0022-3719/13/19/002}
+}
+
+@article{Bray_2007_Statistics,
+ author = {Bray, Alan J. and Dean, David S.},
+ title = {Statistics of Critical Points of {Gaussian} Fields on Large-Dimensional Spaces},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {2007},
+ month = {4},
+ number = {15},
+ volume = {98},
+ pages = {150201},
+ url = {https://doi.org/10.1103%2Fphysrevlett.98.150201},
+ doi = {10.1103/physrevlett.98.150201}
+}
+
+@article{Cartwright_2013_The,
+ author = {Cartwright, Dustin and Sturmfels, Bernd},
+ title = {The number of eigenvalues of a tensor},
+ journal = {Linear Algebra and its Applications},
+ publisher = {Elsevier BV},
+ year = {2013},
+ month = {1},
+ number = {2},
+ volume = {438},
+ pages = {942--952},
+ url = {https://doi.org/10.1016%2Fj.laa.2011.05.040},
+ doi = {10.1016/j.laa.2011.05.040}
+}
+
+@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{Cavagna_1999_Energy,
  author = {Cavagna, Andrea and Garrahan, Juan P. and Giardina, Irene},
  title = {Energy distribution of maxima and minima in a one-dimensional random system},
@@ -12,6 +159,76 @@
  doi = {10.1103/physreve.59.2808}
 }
 
+@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_1995_Thouless-Anderson-Palmer,
+ author = {Crisanti, A. and Sommers, H.-J.},
+ title = {{Thouless}-{Anderson}-{Palmer} Approach to the Spherical p-Spin Spin Glass Model},
+ journal = {Journal de Physique I},
+ publisher = {EDP Sciences},
+ year = {1995},
+ month = {7},
+ number = {7},
+ volume = {5},
+ pages = {805--813},
+ url = {https://doi.org/10.1051%2Fjp1%3A1995164},
+ doi = {10.1051/jp1:1995164}
+}
+
+@article{Cristoforetti_2012_New,
+ author = {Cristoforetti, Marco and Di Renzo, Francesco and Scorzato, Luigi},
+ title = {New approach to the sign problem in quantum field theories: High density {QCD} on a {Lefschetz} thimble},
+ journal = {Physical Review D},
+ publisher = {American Physical Society (APS)},
+ year = {2012},
+ month = {10},
+ number = {7},
+ volume = {86},
+ pages = {074506},
+ url = {https://doi.org/10.1103%2Fphysrevd.86.074506},
+ doi = {10.1103/physrevd.86.074506}
+}
+
+@article{Cugliandolo_1993_Analytical,
+ author = {Cugliandolo, L. F. and Kurchan, J.},
+ title = {Analytical solution of the off-equilibrium dynamics of a long-range spin-glass model},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {1993},
+ month = {7},
+ number = {1},
+ volume = {71},
+ pages = {173--176},
+ url = {https://doi.org/10.1103%2Fphysrevlett.71.173},
+ doi = {10.1103/physrevlett.71.173}
+}
+
+@article{Dyson_1962_A,
+ author = {Dyson, Freeman J.},
+ title = {A {Brownian}-Motion Model for the Eigenvalues of a Random Matrix},
+ journal = {Journal of Mathematical Physics},
+ publisher = {AIP Publishing},
+ year = {1962},
+ month = {11},
+ number = {6},
+ volume = {3},
+ pages = {1191--1198},
+ url = {https://doi.org/10.1063%2F1.1703862},
+ doi = {10.1063/1.1703862}
+}
+
 @book{Forstneric_2017_Stein,
  author = {Forstnerič, Franc},
  title = {{Stein} Manifolds and Holomorphic Mappings},
@@ -26,6 +243,33 @@
  subtitle = {The Homotopy Principle in Complex Analysis}
 }
 
+@article{Fyodorov_2004_Complexity,
+ author = {Fyodorov, Yan V.},
+ title = {Complexity of Random Energy Landscapes, Glass Transition, and Absolute Value of the Spectral Determinant of Random Matrices},
+ journal = {Physical Review Letters},
+ publisher = {American Physical Society (APS)},
+ year = {2004},
+ month = {6},
+ number = {24},
+ volume = {92},
+ pages = {240601},
+ url = {https://doi.org/10.1103%2Fphysrevlett.92.240601},
+ doi = {10.1103/physrevlett.92.240601}
+}
+
+@article{Kac_1943_On,
+ author = {Kac, M.},
+ title = {On the average number of real roots of a random algebraic equation},
+ journal = {Bulletin of the American Mathematical Society},
+ publisher = {American Mathematical Society},
+ year = {1943},
+ month = {4},
+ number = {4},
+ volume = {49},
+ pages = {314--320},
+ url = {https://projecteuclid.org:443/euclid.bams/1183505112}
+}
+
 @article{Kent-Dobias_2021_Complex,
  author = {Kent-Dobias, Jaron and Kurchan, Jorge},
  title = {Complex complex landscapes},
@@ -40,6 +284,41 @@
  doi = {10.1103/physrevresearch.3.023064}
 }
 
+@book{Livan_2018_Introduction,
+ author = {Livan, Giacomo and Novaes, Marcel and Vivo, Pierpaolo},
+ title = {Introduction to Random Matrices},
+ publisher = {Springer International Publishing},
+ year = {2018},
+ volume = {26},
+ url = {https://doi.org/10.1007%2F978-3-319-70885-0},
+ doi = {10.1007/978-3-319-70885-0},
+ series = {SpringerBriefs in Mathematical Physics},
+ subtitle = {Theory and Practice}
+}
+
+@article{Manschot_2012_From,
+ author = {Manschot, Jan and Pioline, Boris and Sen, Ashoke},
+ title = {From black holes to quivers},
+ journal = {Journal of High Energy Physics},
+ publisher = {Springer Science and Business Media LLC},
+ year = {2012},
+ month = {11},
+ volume = {2012},
+ pages = {23},
+ url = {https://doi.org/10.1007%2Fjhep11%282012%29023},
+ doi = {10.1007/jhep11(2012)023}
+}
+
+@book{Mezard_2009_Information,
+ author = {Mézard, Marc and Montanari, Andrea},
+ title = {Information, physics, and computation},
+ publisher = {Oxford University Press},
+ year = {2009},
+ address = {Great Clarendon Street, Oxford},
+ isbn = {9780198570837},
+ series = {Oxford Graduate Texts}
+}
+
 @book{Morrow_2006_Complex,
  author = {Morrow, James and Kodaira, Kunihiko},
  title = {Complex manifolds},
@@ -50,9 +329,91 @@
  isbn = {9780821840559}
 }
 
+@article{Nguyen_2014_The,
+ author = {Nguyen, Hoi H. and O'Rourke, Sean},
+ title = {The Elliptic Law},
+ journal = {International Mathematics Research Notices},
+ publisher = {Oxford University Press (OUP)},
+ year = {2014},
+ month = {10},
+ number = {17},
+ volume = {2015},
+ pages = {7620--7689},
+ url = {https://doi.org/10.1093%2Fimrn%2Frnu174},
+ doi = {10.1093/imrn/rnu174}
+}
+
+@article{Rice_1939_The,
+ author = {Rice, S. O.},
+ title = {The Distribution of the Maxima of a Random Curve},
+ journal = {American Journal of Mathematics},
+ publisher = {JSTOR},
+ year = {1939},
+ month = {4},
+ number = {2},
+ volume = {61},
+ pages = {409},
+ url = {https://doi.org/10.2307%2F2371510},
+ doi = {10.2307/2371510}
+}
+
+@inproceedings{Scorzato_2016_The,
+ author = {Scorzato, Luigi},
+ title = {The {Lefschetz} thimble and the sign problem},
+ publisher = {Sissa Medialab},
+ year = {2016},
+ month = {7},
+ volume = {251},
+ url = {https://doi.org/10.22323%2F1.251.0016},
+ doi = {10.22323/1.251.0016},
+ booktitle = {Proceedings of The 33rd International Symposium on Lattice Field Theory ({LATTICE} 2015)},
+ series = {Proceedings of Science}
+}
+
+@article{Tanizaki_2017_Gradient,
+ author = {Tanizaki, Yuya and Nishimura, Hiromichi and Verbaarschot, Jacobus J. M.},
+ title = {Gradient flows without blow-up for {Lefschetz} thimbles},
+ journal = {Journal of High Energy Physics},
+ publisher = {Springer Science and Business Media LLC},
+ year = {2017},
+ month = {10},
+ number = {10},
+ volume = {2017},
+ pages = {100},
+ url = {https://doi.org/10.1007%2Fjhep10%282017%29100},
+ doi = {10.1007/jhep10(2017)100}
+}
+
+@article{Weyl_1912_Das,
+ author = {Weyl, Hermann},
+ title = {Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung)},
+ journal = {Mathematische Annalen},
+ publisher = {Springer Science and Business Media LLC},
+ year = {1912},
+ month = {12},
+ number = {4},
+ volume = {71},
+ pages = {441--479},
+ url = {https://doi.org/10.1007%2Fbf01456804},
+ doi = {10.1007/bf01456804}
+}
+
+@article{Witten_2010_A,
+ author = {Witten, Edward},
+ title = {A new look at the path integral of quantum mechanics},
+ journal = {Surveys in Differential Geometry},
+ publisher = {International Press of Boston},
+ year = {2010},
+ number = {1},
+ volume = {15},
+ pages = {345--420},
+ url = {https://doi.org/10.4310%2Fsdg.2010.v15.n1.a11},
+ doi = {10.4310/sdg.2010.v15.n1.a11}
+}
+
 @incollection{Witten_2011_Analytic,
  author = {Witten, Edward},
- title = {Analytic continuation of Chern-Simons theory},
+ title = {Analytic continuation of {Chern}-{Simons} theory},
  publisher = {American Mathematical Society},
  year = {2011},
  month = {7},
@@ -60,7 +421,7 @@
  pages = {347--446},
  url = {https://doi.org/10.1090%2Famsip%2F050%2F19},
  doi = {10.1090/amsip/050/19},
- booktitle = {Chern-Simons Gauge Theory: 20 Years After},
+ booktitle = {{Chern}-{Simons} Gauge Theory: 20 Years After},
  editor = {Andersen, Jørgen E. and Boden, Hans U. and Hahn, Atle and Himpel, Benjamin},
  series = {AMS/IP Studies in Advanced Mathematics}
 }
diff --git a/stokes.tex b/stokes.tex
index a03f901..61d8880 100644
--- a/stokes.tex
+++ b/stokes.tex
@@ -15,6 +15,9 @@
 \begin{document}
 
 \newcommand\eqref[1]{\eref{#1}}
+\makeatletter
+\renewcommand\tableofcontents{\@starttoc{toc}}
+\makeatother
 
 \title{Analytic continuation over complex landscapes}
 
@@ -44,6 +47,8 @@
 
 \maketitle
 
+\tableofcontents
+
 \section{Preamble}
 
 Complex landscapes are basically functions of many variables having many minima and, inevitably,
@@ -806,6 +811,122 @@ $\det U=i^k$. As the argument of $\beta$ is changed, we know how the eigenvector
   =Z(\beta^*)
 \end{eqnarray}
 
+\section{The ensemble of symmetric complex-normal matrices}
+
+The Hessian $\partial\partial H=\partial\partial H_0-p\epsilon I$ is equal to
+the unconstrained Hessian with a constant added to its diagonal. The eigenvalue
+distribution $\rho$ is therefore related to the unconstrained distribution
+$\rho_0$ by a similar shift: $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The
+Hessian of the unconstrained Hamiltonian is
+\begin{equation} \label{eq:bare.hessian}
+  \partial_i\partial_jH_0
+  =\frac{p(p-1)}{p!}\sum_{k_1\cdots k_{p-2}}^NJ_{ijk_1\cdots k_{p-2}}z_{k_1}\cdots z_{k_{p-2}},
+\end{equation}
+which makes its ensemble that of Gaussian complex symmetric matrices, when the
+anomalous direction normal to the constraint surface is neglected. Given its variances
+$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)r^{p-2}/2N$ and
+$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is
+constant inside the ellipse
+\begin{equation} \label{eq:ellipse}
+  \left(\frac{\operatorname{Re}(\lambda e^{i\theta})}{r^{p-2}+|\kappa|}\right)^2+
+  \left(\frac{\operatorname{Im}(\lambda e^{i\theta})}{r^{p-2}-|\kappa|}\right)^2
+  <\frac{p(p-1)}{2r^{p-2}}
+\end{equation}
+where $\theta=\frac12\arg\kappa$ \cite{Nguyen_2014_The}. The eigenvalue
+spectrum of $\partial\partial H$ is therefore constant inside the same ellipse
+translated so that its center lies at $-p\epsilon$.  Examples of these
+distributions are shown in the insets of Fig.~\ref{fig:spectra}.
+
+The eigenvalue spectrum of the Hessian of the real part is not the
+spectrum $\rho(\lambda)$ of $\partial\partial H$, but instead the
+square-root eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$;
+in other words, the singular value spectrum $\rho(\sigma)$ of $\partial\partial
+H$. When $\kappa=0$ and the elements of $J$ are standard complex normal, this
+is a complex Wishart distribution. For $\kappa\neq0$ the problem changes, and
+to our knowledge a closed form is not in the literature.  We have worked out an
+implicit form for the singular value spectrum using the replica method.
+
+Introducing replicas to bring the partition function into the numerator of the
+Green function \cite{Livan_2018_Introduction} gives
+\begin{equation} \label{eq:green.replicas}
+  G(\sigma)=\lim_{n\to0}\int d\zeta\,d\zeta^*\,(\zeta_i^{(0)})^*\zeta_i^{(0)}
+    \exp\left\{
+    \frac12\sum_\alpha^n\left[(\zeta_i^{(\alpha)})^*\zeta_i^{(\alpha)}\sigma
+      -\operatorname{Re}\left(\zeta_i^{(\alpha)}\zeta_j^{(\alpha)}\partial_i\partial_jH\right)
+    \right]
+  \right\},
+\end{equation}
+with sums taken over repeated Latin indices. The average is then made over
+$J$ and Hubbard--Stratonovich is used to change variables to the replica matrices
+$N\alpha_{\alpha\beta}=(\zeta^{(\alpha)})^\dagger\zeta^{(\beta)}$ and
+$N\chi_{\alpha\beta}=(\zeta^{(\alpha)})^T\zeta^{(\beta)}$, and a series of
+replica vectors. The replica-symmetric ansatz leaves all replica vectors
+zero, and $\alpha_{\alpha\beta}=\alpha_0\delta_{\alpha\beta}$,
+$\chi_{\alpha\beta}=\chi_0\delta_{\alpha\beta}$. The result is
+\begin{equation}\label{eq:green.saddle}
+  \overline G(\sigma)=N\lim_{n\to0}\int d\alpha_0\,d\chi_0\,d\chi_0^*\,\alpha_0
+  \exp\left\{nN\left[
+    1+\frac{p(p-1)}{16}r^{p-2}\alpha_0^2-\frac{\alpha_0\sigma}2+\frac12\log(\alpha_0^2-|\chi_0|^2)
+    +\frac p4\operatorname{Re}\left(\frac{(p-1)}8\kappa^*\chi_0^2-\epsilon^*\chi_0\right)
+  \right]\right\}.
+\end{equation}
+
+\begin{figure}
+  \centering
+
+  \includegraphics{figs/spectra_0.0.pdf}
+  \includegraphics{figs/spectra_0.5.pdf}\\
+  \includegraphics{figs/spectra_1.0.pdf}
+  \includegraphics{figs/spectra_1.5.pdf}
+
+  \caption{
+    Eigenvalue and singular value spectra of the Hessian $\partial\partial H$
+    of the $3$-spin model with $\kappa=\frac34e^{-i3\pi/4}$. Pictured
+    distributions are for critical points at `radius' $r=\sqrt{5/4}$ and with
+    energy per spin (a) $\epsilon=0$, (b)
+    $\epsilon=-\frac12|\epsilon_{\mathrm{th}}|$, (c)
+    $\epsilon=-|\epsilon_{\mathrm{th}}|$, and (d)
+    $\epsilon=-\frac32|\epsilon_{\mathrm{th}}|$. The shaded region of each
+    inset shows the support of the eigenvalue distribution \eqref{eq:ellipse}.
+    The solid line on each plot shows the distribution of singular values
+    \eqref{eq:spectral.density}, while the overlaid histogram shows the
+    empirical distribution from $2^{10}\times2^{10}$ complex normal matrices
+    with the same covariance and diagonal shift as $\partial\partial H$.
+  } \label{fig:spectra}
+\end{figure}
+
+The argument of the exponential has several saddles. The solutions $\alpha_0$
+are the roots of a sixth-order polynomial, and the root with the smallest value
+of $\operatorname{Re}\alpha_0$ gives the correct solution in all the cases we
+studied. A detailed analysis of the saddle point integration is needed to
+understand why this is so. Evaluated at such a solution, the density of
+singular values follows from the jump across the cut, or
+\begin{equation} \label{eq:spectral.density}
+  \rho(\sigma)=\frac1{i\pi N}\left(
+    \lim_{\operatorname{Im}\sigma\to0^+}\overline G(\sigma)
+    -\lim_{\operatorname{Im}\sigma\to0^-}\overline G(\sigma)
+  \right)
+\end{equation}
+Examples can be seen in Fig.~\ref{fig:spectra} compared with numeric
+experiments.
+
+The formation of a gap in the singular value spectrum naturally corresponds to
+the origin leaving the support of the eigenvalue spectrum.  Weyl's theorem
+requires that the product over the norm of all eigenvalues must not be greater
+than the product over all singular values \cite{Weyl_1912_Das}.  Therefore, the
+absence of zero eigenvalues implies the absence of zero singular values. The
+determination of the threshold energy---the energy at which the distribution
+of singular values becomes gapped---is reduced to the geometry problem of
+determining when the boundary of the ellipse defined in \eqref{eq:ellipse}
+intersects the origin, and yields
+\begin{equation} \label{eq:threshold.energy}
+  |\epsilon_{\mathrm{th}}|^2
+  =\frac{p-1}{2p}\frac{(1-|\delta|^2)^2r^{p-2}}
+  {1+|\delta|^2-2|\delta|\cos(\arg\kappa+2\arg\epsilon)}
+\end{equation}
+for $\delta=\kappa r^{-(p-2)}$. Notice that the threshold depends on both the
+energy per spin $\epsilon$ on the `radius' $r$ of the saddle.
+
 \section{The \textit{p}-spin spherical models}
 
 The $p$-spin spherical models are statistical mechanics models defined by the
@@ -830,8 +951,6 @@ this manifold is given by
 \end{equation}
 where indeed $Pz=z-z|z|^2/|z|^2=0$ and $Pz'=z'$ for any $z'$ orthogonal to $z$.
 
-To find critical points, we use the method of Lagrange multipliers. Introducing $\mu\in\mathbb C$,
-
 \subsection{2-spin}
 
 The Hamiltonian of the $2$-spin model is defined for $z\in\mathbb C^N$ by
@@ -901,7 +1020,15 @@ imaginary energy join.
 
 \begin{eqnarray}
   Z(\beta)
-  &=\int_{S^{N-1}}dx\,e^{-\beta H(x)}
+  &=\int_{S^{N-1}}ds\,e^{-\beta H(s)}
+  =\sum_{\sigma\in\Sigma_0}n_\sigma\int_{\mathcal J_\sigma}ds\,e^{-\beta H_2(s)} \\
+  &\simeq\sum_{k=0}^D2n_ki^k\left(\frac{2\pi}\beta\right)^{D/2}e^{-\beta H_2(s_\sigma)}\prod_{\lambda_0>0}\lambda_0^{-1/2} \\
+  &=\sum_{k=0}^D2n_ki^k\left(\frac{2\pi}\beta\right)^{D/2}e^{-N\beta\epsilon_k}\prod_{\ell\neq k}\frac12|\epsilon_k-\epsilon_\ell|
+\end{eqnarray}
+
+\begin{eqnarray}
+  Z(\beta)
+  &=\int_{S^{N-1}}ds\,e^{-\beta H(s)}
   =\int_{\mathbb R^N}dx\,\delta(x^2-N)e^{-\beta H(x)} \\
   &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12\beta x^TJx-\lambda(x^Tx-N)} \\
   &=\frac1{2\pi}\int_{\mathbb R^N}dx\,d\lambda\,e^{-\frac12x^T(\beta J+\lambda I)x+\lambda N} \\
@@ -913,6 +1040,208 @@ imaginary energy join.
 
 \subsection{Pure \textit{p}-spin}
 
+Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also
+one of $\operatorname{Im}H$, and therefore of $H$ itself. Writing $z=x+iy$ for
+$x,y\in\mathbb R^N$, $\operatorname{Re}H$ can be considered a real-valued
+function of $2N$ real variables. The number of critical points of $H$ is thus given by the
+usual Kac--Rice formula applied to $\operatorname{Re}H$:
+\begin{equation} \label{eq:real.kac-rice}
+  \mathcal N(\kappa,\epsilon,R)
+    = \int dx\,dy\,\delta(\partial_x\operatorname{Re}H)\delta(\partial_y\operatorname{Re}H)
+            \left|\det\operatorname{Hess}_{x,y}\operatorname{Re}H\right|.
+\end{equation}
+This expression is to be averaged over $J$ to give the complexity $\Sigma$ as
+$N \Sigma= \overline{\log\mathcal N}$, a calculation that involves the replica
+trick. Based on the experience from similar problems \cite{Castellani_2005_Spin-glass},  the
+\emph{annealed approximation} $N \Sigma \sim \log \overline{ \mathcal N}$ is
+expected to be exact wherever the complexity is positive.
+
+The Cauchy--Riemann equations may be used to write \eqref{eq:real.kac-rice} in
+a manifestly complex way.  With the Wirtinger derivative
+$\partial\equiv\frac12(\partial_x-i\partial_y)$, one can write
+$\partial_x\operatorname{Re}H=\operatorname{Re}\partial H$ and
+$\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying these
+transformations through, one finds
+\begin{equation} \label{eq:complex.kac-rice}
+  \mathcal N(\kappa,\epsilon,r)
+    = \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H)
+    |\det\operatorname{Hess}H|^2.
+\end{equation}
+This gives three equivalent expressions for the determinant of the Hessian: as
+that of a $2N\times 2N$ real symmetric matrix, that of the $N\times N$ Hermitian
+matrix $(\partial\partial H)^\dagger\partial\partial H$, or the norm squared of
+that of the $N\times N$ complex symmetric matrix $\partial\partial H$.
+
+These equivalences belie a deeper connection between the spectra of the
+corresponding matrices. Each positive eigenvalue of the real matrix has a
+negative partner. For each such pair $\pm\lambda$, $\lambda^2$ is an eigenvalue
+of the Hermitian matrix and $|\lambda|$ is a \emph{singular value} of the
+complex symmetric matrix. The distribution of positive eigenvalues of the
+Hessian is therefore the same as the distribution of singular values of
+$\partial\partial H$, or the distribution of square-rooted eigenvalues of
+$(\partial\partial H)^\dagger\partial\partial H$.
+
+A useful property of the Gaussian $J$ is that gradient and Hessian at fixed
+energy $\epsilon$ are statistically independent \cite{Bray_2007_Statistics,
+Fyodorov_2004_Complexity}, so that the $\delta$-functions and the Hessian may
+be averaged independently. First we shall compute the spectrum of the Hessian,
+which can in turn be used to compute the determinant. Then we will treat the
+$\delta$-functions and the resulting saddle point equations. The results of
+these calculations begin around \eqref{eq:bezout}.
+
+
+We will now address the $\delta$-functions of \eqref{eq:complex.kac-rice}.
+These are converted to exponentials by the introduction of auxiliary fields
+$\hat z=\hat x+i\hat y$.  The average over $J$ can then be performed. A
+generalized Hubbard--Stratonovich allows a change of variables from the $4N$
+original and auxiliary fields to eight bilinears defined by $Nr=z^\dagger z$,
+$N\hat r=\hat z^\dagger\hat z$, $Na=\hat z^\dagger z$, $Nb=\hat z^Tz$, and
+$N\hat c=\hat z^T\hat z$ (and their conjugates). The result, to leading order
+in $N$, is
+\begin{equation} \label{eq:saddle}
+    \overline{\mathcal N}(\kappa,\epsilon,R)
+        = \int dr\,d\hat r\,da\,da^*db\,db^*d\hat c\,d\hat c^*e^{Nf(r,\hat r,a,b,\hat c)},
+\end{equation}
+where the argument of the exponential is
+\begin{equation}
+  f=2+\frac12\log\det\frac12\left[\matrix{
+    1 & r & b & a \cr
+    r & 1 & a^* & b^* \cr
+    b & a^* & \hat c & \hat r \cr
+    a & b^* & \hat r & \hat c^*
+  }\right]
+  +\int d\lambda\,d\lambda^*\rho(\lambda)\log|\lambda|^2
+  +p\operatorname{Re}\left\{
+    \frac18\left[\hat rr^{p-1}+(p-1)|b|^2r^{p-2}+\kappa(\hat c^*+(p-1)a^2)\right]-\epsilon a
+  \right\}.
+\end{equation}
+The spectrum $\rho$ is given in \eqref{eq:ellipse} and is dependant on $r$ alone. This function has an
+extremum in $\hat r$, $a$, $b$, and $\hat c$ at which its value is
+\begin{equation} \label{eq:free.energy.a}
+    f=1+\frac12\log\left(\frac4{p^2}\frac{r^2-1}{r^{2(p-1)}-|\kappa|^2}\right)+\int d\lambda\,\rho(\lambda)\log|\lambda|^2
+    -2C_+[\operatorname{Re}(\epsilon e^{-i\theta})]^2-2C_-[\operatorname{Im}(\epsilon e^{-i\theta})]^2,
+\end{equation}
+where $\theta=\frac12\arg\kappa$ and
+\begin{equation}
+  C_{\pm}=\frac{r^p(1+p(r^2-1))\mp r^2|\kappa|}{r^{2p}\pm(p-1)r^p(r^2-1)|\kappa|-r^2|\kappa|^2}.
+\end{equation}
+Notice that level sets of $f$ in energy $\epsilon$ also give ellipses, but of
+different form from the ellipse in \eqref{eq:ellipse}.
+
+This expression is maximized for $r=R$, its value at the boundary, for
+all values of $\kappa$ and $\epsilon$. Evaluating the complexity at this
+saddle, in the limit of unbounded spins, gives
+\begin{equation} \label{eq:bezout}
+  \lim_{R\to\infty}\log\overline{\mathcal N}(\kappa,\epsilon,R)
+  =N\log(p-1).
+\end{equation}
+This is, to leading order, precisely the Bézout bound, the maximum number of
+solutions to $N$ equations of degree $p-1$ \cite{Bezout_1779_Theorie}. That we
+saturate this bound is perhaps not surprising, since the coefficients of our
+polynomial equations \eqref{eq:polynomial} are complex and have no symmetries.
+Reaching Bézout in \eqref{eq:bezout} is not our main result, but it provides a
+good check.  Analogous asymptotic scaling has been found for the number of pure
+Higgs states in supersymmetric quiver theories \cite{Manschot_2012_From}.
+
+\begin{figure}[htpb]
+  \centering
+  \includegraphics{figs/complexity.pdf}
+  \caption{
+    The complexity of the 3-spin model as a function of the maximum `radius'
+    $R$ at zero energy and several values of $\kappa$. The dashed line shows
+    $\frac12\log(p-1)$, while the dotted shows $\log(p-1)$.
+  } \label{fig:complexity}
+\end{figure}
+
+For finite $R$, everything is analytically tractable at $\epsilon=0$:
+\begin{equation} \label{eq:complexity.zero.energy}
+  \Sigma(\kappa,0,R)
+  =\log(p-1)-\frac12\log\left(\frac{1-|\kappa|^2R^{-4(p-1)}}{1-R^{-4}}\right).
+\end{equation}
+This is plotted as a function of $R$ for several values of $\kappa$ in
+Fig.~\ref{fig:complexity}. For any $|\kappa|<1$, the complexity goes to
+negative infinity as $R\to1$, i.e., as the spins are restricted to the reals.
+This is natural, since volume of configuration space vanishes in this limit
+like $(R^2-1)^N$. However, when the result is analytically continued to
+$\kappa=1$ (which corresponds to real $J$) something novel occurs: the
+complexity has a finite value at $R=1$. This implies a $\delta$-function
+density of critical points on the $r=1$ (or $y=0$) boundary.  The number of
+critical points contained there is
+\begin{equation}
+  \lim_{R\to1}\lim_{\kappa\to1}\log\overline{\mathcal N}(\kappa,0,R)
+  = \frac12N\log(p-1),
+\end{equation}
+half of \eqref{eq:bezout} and corresponding precisely to the number of critical
+points of the real $p$-spin model. (Note the role of conjugation symmetry,
+already underlined in \cite{Bogomolny_1992_Distribution}.) The full
+$\epsilon$-dependence of the real $p$-spin is recovered by this limit as
+$\epsilon$ is varied.
+
+\begin{figure}[b]
+  \centering
+  \includegraphics{figs/desert.pdf}
+  \caption{
+    The value of bounding `radius' $R$ for which $\Sigma(\kappa,\epsilon,R)=0$ as a
+    function of (real) energy per spin $\epsilon$  for the 3-spin model at
+    several values of $\kappa$. Above each line the complexity is positive and
+    critical points proliferate, while below it the complexity is negative and
+    critical points are exponentially suppressed. The dotted black lines show
+    the location of the ground and highest exited state energies for the real
+    3-spin model.
+  } \label{fig:desert}
+\end{figure}
+
+In the thermodynamic limit, \eqref{eq:complexity.zero.energy} implies that most
+critical points are concentrated at infinite radius $r$. For finite $N$ the
+average radius of critical points is likewise finite. By differentiating
+$\overline{\mathcal N}$ with respect to $R$ and normalizing, one obtains the
+distribution of critical points as a function of $r$. This yields an average
+radius proportional to $N^{1/4}$.  One therefore expects typical critical
+points to have a norm that grows modestly with system size.
+
+These qualitative features carry over to nonzero $\epsilon$. In
+Fig.~\ref{fig:desert} we show that for $\kappa<1$ there is always a gap in $r$
+close to one in which solutions are exponentially suppressed. When
+$\kappa=1$---the analytic continuation to the real computation---the situation
+is more interesting. In the range of energies where there are real solutions
+this gap closes, which is only possible if the density of solutions diverges at
+$r=1$. Outside this range, around `deep' real energies where real solutions are
+exponentially suppressed, the gap remains. A moment's thought tells us that
+this is necessary: otherwise a small perturbation of the $J$s could produce
+an unusually deep solution to the real problem, in a region where this should
+not happen.
+
+\begin{figure}[t]
+  \centering
+
+  \includegraphics{figs/threshold_2.000.pdf}
+  \includegraphics{figs/threshold_1.325.pdf} \\
+  \includegraphics{figs/threshold_1.125.pdf}
+  \includegraphics{figs/threshold_1.000.pdf}
+
+  \caption{
+    Energies at which states exist (green shaded region) and threshold energies
+    (black solid line) for the 3-spin model with
+    $\kappa=\frac34e^{-i3\pi/4}$ and (a) $r=\sqrt2$, (b) $r=\sqrt{1.325}$, (c) $r=\sqrt{1.125}$,
+    and (d) $r=1$. No shaded region is shown in (d) because no states exist at
+    any energy.
+  } \label{fig:eggs}
+\end{figure}
+
+The relationship between the threshold and ground, or extremal, state energies
+is richer than in the real case.  In Fig.~\ref{fig:eggs} these are shown in the
+complex-$\epsilon$ plane for several examples. Depending on the parameters, the
+threshold might have a smaller or larger magnitude than the extremal state, or
+cross as a function of complex argument.  For sufficiently large $r$ the
+threshold is always at a larger magnitude. If this were to happen in the real
+case, it would likely imply our replica symmetric computation were unstable,
+since having a ground state above the threshold implies a ground state Hessian
+with many negative eigenvalues, a contradiction. However, this is not an
+contradiction in the complex case, where the energy is not bounded from below.
+The relationship between the threshold, i.e., where the gap appears, and the
+dynamics of, e.g., a minimization algorithm, deformed integration cycle, or
+physical dynamics, are a problem we hope to address in future work.
+
 \begin{equation}
   H_p=\frac1{p!}\sum_{i_1\cdots i_p}J_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p}
 \end{equation}