Merge branch 'master' of https://git.overleaf.com/5fcce4736e7f601ffb7e1484

1 files changed, 20 insertions, 11 deletions

diff --git a/bezout.tex b/bezout.tex
index 6873e58..407e58d 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -41,28 +41,28 @@ Spin-glasses have long been considered the paradigm of `complex landscapes' of m
 includes Neural Networks and optimization problems, most notably  Constraint Satisfaction ones.
 The most tractable family of these  are the mean-field spherical p-spin models defined by the energy:
 \begin{equation} \label{eq:bare.hamiltonian}
-  E = \sum_p \frac{c_p}{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
+  H_o = \sum_p \frac{c_p}{p!}\sum_{i_1\cdots i_p}^NJ_{i_1\cdots i_p}z_{i_1}\cdots z_{i_p},
 \end{equation}
 where the $J_{i_1\cdots i_p}$ are real Gaussian variables and the $z_i$ are real and constrained
-to a sphere $\sum_i z_i^2=N$.
+to a sphere $\sum_i z_i^2=N$. If there is a single  term of a given $p$, this is known as the `pure $p$-spin' model, the case we shall study here.
 
 This problem has been attacked from several angles: the replica trick to compute the Boltzmann-Gibbs distribution,
-a Kac-Rice \cite{Kac,Fyodorov} procedure (similar to the Fadeev-Popov inte to compute the number of saddle-points of the energy function, and the gradient-descent -- or more generally Langevin -- dynamics staring from a high-energy configuration.
+a Kac-Rice \cite{Kac,Fyodorov} procedure (similar to the Fadeev-Popov integral)  to compute the number of saddle-points of the energy function, and the gradient-descent -- or more generally Langevin -- dynamics staring from a high-energy configuration.
 Thanks to the relative simplicity of the energy, all these approaches are possible analytically in the large $N$ limit.
 
-In th
-where $z\in\mathbb C^N$ is constrained by $z^2=N$ and $J$ is a symmetric tensor
+In this paper we shall extend the study to the case  where $z\in\mathbb C^N$ are  and $J$ is a symmetric tensor
 whose elements are complex normal with $\langle|J|^2\rangle=p!/2N^{p-1}$ and
-$\langle J^2\rangle=\kappa\langle|J|^2\rangle$ for complex parameter
-$|\kappa|<1$. The constraint is enforced using the method of Lagrange
+$\langle J^2\rangle=\kappa\langle|J|^2\rangle$ for complex parameter $|\kappa|<1$. 
+
+It is enforced using the method of Lagrange
 multipliers: introducing the $\epsilon\in\mathbb C$, this gives
 \begin{equation} \label{eq:constrained.hamiltonian}
   H = H_0+\frac p2\epsilon\left(N-\sum_i^Nz_i^2\right).
 \end{equation}
-At any critical point $\epsilon=H/N$, the average energy.
+It is easy to see that {\em for a pure $p$ spin}, at  any critical point $\epsilon=H/N$, the average energy.
 
-When compared with $z^*z=N$, the constraint $z^2=N$ may seem an unnatural
-extension of the real $p$-spin spherical model. However, a model with this
+For the  constraint we shall choose here $z^2=N$, rather than $|z^2|=N$, in order to preserve the holomorphic nature
+of the functions.. In addition, the  
 nonholomorphic spherical constraint has a disturbing lack of critical points
 nearly everywhere, since $0=\partial^* H=-p\epsilon z$ is only
 satisfied for $\epsilon=0$, as $z=0$ is forbidden by the constraint.
@@ -81,7 +81,16 @@ points it has is given by the usual Kac--Rice formula:
         \partial_y\partial_x\mathop{\mathrm{Re}}H & \partial_y\partial_y\mathop{\mathrm{Re}}H
       \end{bmatrix}\right|.
 \end{equation}
-The Cauchy--Riemann relations can be used to simplify this. Using the Wirtinger
+The Cauchy--Riemann relations imply that the matrix is of the form:
+\begin{equation} \label{eq:real.kac-rice1}
+      \left|\det\begin{bmatrix}
+        A & B \\
+        B & -A
+      \end{bmatrix}\right|.
+\end{equation}
+
+
+Using the Wirtinger
 derivative $\partial=\partial_x-i\partial_y$, one can write
 $\partial_x\mathop{\mathrm{Re}}H=\mathop{\mathrm{Re}}\partial H$ and
 $\partial_y\mathop{\mathrm{Re}}H=-\mathop{\mathrm{Im}}\partial H$. With similar