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-rw-r--r--bezout.tex34
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