Update on Overleaf.

1 files changed, 21 insertions, 13 deletions

diff --git a/bezout.tex b/bezout.tex
index 407e58d..87b0a9f 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -52,8 +52,23 @@ Thanks to the relative simplicity of the energy, all these approaches are possib
 
 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 $|\kappa|<1$. 
+$\langle J^2\rangle=\kappa\langle|J|^2\rangle$ for complex parameter $|\kappa|<1$. The constraint becomes $z^2=N$.
 
+The motivations for this paper are of two types. On the practical side, there are situations in which  complex variables
+have in a disorder problem appear naturally: such is the case in which they are {\em phases}, as in random laser problems \cite{Leuzzi-Crisanti, etc}. Another problem where a Hamiltonian very close to ours has been proposed is the Quiver Hamiltonians \cite{Anninos-Anous-Denef} modeling Black Hole horizons in the zero-temperature limit.  
+
+There is however a more fundamental reason for this study: we know from experience that extending a problem to the complex plane often uncovers an underlying simplicity that is hidden in the purely real case. Consider, for example, the procedure of starting from a simple, known Hamiltonian $H_{00}$ and studying $\lambda H_{00} + (1-\lambda H_{0} )$, evolving adiabatically from $\lambda=0$ to $\lambda=1$, as is familiar from quantum annealing. The $H_{00}$ is a polynomial of degree $p$  chosen to have simple, known roots. Because we are working in
+complex variables, and the roots are simple all the way (we shall confirm this), we may follow a root from $\lambda=0$ to $\lambda=1$. With real 
+variables minima of functions appear and disappear, and this procedure is not possible. The same idea may be implemented by 
+performing diffusion in the $J$'s, and following the roots, in complete analogy with Dyson's stochastic dynamics.
+
+
+Let us go back to our model.
+For the  constraint we  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.
 It is enforced using the method of Lagrange
 multipliers: introducing the $\epsilon\in\mathbb C$, this gives
 \begin{equation} \label{eq:constrained.hamiltonian}
@@ -61,11 +76,6 @@ multipliers: introducing the $\epsilon\in\mathbb C$, this gives
 \end{equation}
 It is easy to see that {\em for a pure $p$ spin}, at  any critical point $\epsilon=H/N$, the average energy.
 
-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.
 
 Since $H$ is holomorphic, a point is a critical point of its real part if and
 only if it is also a critical point of its imaginary part. The number of
@@ -83,13 +93,11 @@ points it has is given by the usual Kac--Rice formula:
 \end{equation}
 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}
-
-
+      \begin{bmatrix}
+        \bar A & \bar B \\
+        \bar B & -\bar A
+      \end{bmatrix}
+\end{equation}with $\bar A$ and $\bar B$  Gaussian real symmetric matrices, {\em correlated}, as we shall see..
 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