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-rw-r--r--bezout.tex40
1 files changed, 21 insertions, 19 deletions
diff --git a/bezout.tex b/bezout.tex
index 67a764b..57e48ae 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -108,9 +108,9 @@ z$ would only be satisfied for $\epsilon=0$.
The critical points are given by the solutions to the set of equations
\begin{equation}
- \frac{p}{p!}\sum_{j_2\cdots j_p}^NJ_{ij_2\cdots j_p}z_{j_2}\cdots z_{j_p} = \epsilon z_i
+ \frac{p}{p!}\sum_{j_1\cdots j_{p-1}}^NJ_{ij_1\cdots j_{p-1}}z_{j_1}\cdots z_{j_{p-1}} = \epsilon z_i
\end{equation}
-for all $i=\{1,\ldots,N\}$, which for given $\epsilon$ are a set of $N$
+for all $i=\{1,\ldots,N\}$, which for given $\epsilon$ is a set of $N$
equations of degree $p-1$, to which one must add the constraint. In this sense
this study also provides a complement to the work on the distribution of zeroes
of random polynomials \cite{Bogomolny_1992_Distribution}, which are for $N=1$
@@ -143,7 +143,7 @@ $\partial_y\operatorname{Re}H=-\operatorname{Im}\partial H$. Carrying
these transformations through, we have
\begin{equation} \label{eq:complex.kac-rice}
\begin{aligned}
- \mathcal N_J&(\kappa,\epsilon)
+ &\mathcal N_J(\kappa,\epsilon)
= \int dx\,dy\,\delta(\operatorname{Re}\partial H)\delta(\operatorname{Im}\partial H) \\
&\qquad\qquad\qquad\times\left|\det\begin{bmatrix}
\operatorname{Re}\partial\partial H & -\operatorname{Im}\partial\partial H \\
@@ -169,10 +169,10 @@ singular values of $\partial\partial H$, the distribution of square-rooted
eigenvalues of $(\partial\partial H)^\dagger\partial\partial H$.
The expression \eqref{eq:complex.kac-rice} is to be averaged over the $J$'s as
-$N \Sigma= \overline{\ln \mathcal N_J} = \int dJ \; \ln N_J$, a calculation
+$N \Sigma= \overline{\log\mathcal N} = \int dJ \, \log \mathcal N_J$, a calculation
that involves the replica trick. In most the parameter-space that we shall
-study here, the {\em annealed approximation} $N \Sigma \sim \ln \overline{
-\mathcal N_J} = \ln \int dJ \; N_J$ is exact.
+study here, the {\em annealed approximation} $N \Sigma \sim \log \overline{
+\mathcal N} = \log\int dJ \, \mathcal N_J$ is exact.
A useful property of the Gaussian distributions is that gradient and Hessian
for given $\epsilon$ may be seen to be independent \cite{Bray_2007_Statistics,
@@ -256,12 +256,14 @@ that of the same ellipse whose 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 the one we need for our Kac--Rice formula. It is different from the spectrum $\partial\partial H$, but rather equivalent to the
-square-root eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$, in other words, the
-singular value spectrum of $\partial\partial H$. When $\kappa=0$ and the
-elements of $J$ are standard complex normal, this corresponds to 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 eigenvalue spectrum of the Hessian of the real part is the one we need for
+our Kac--Rice formula. It is different from the spectrum $\partial\partial H$,
+but rather equivalent to the square-root eigenvalue spectrum of
+$(\partial\partial H)^\dagger\partial\partial H$, in other words, the singular
+value spectrum of $\partial\partial H$. When $\kappa=0$ and the elements of $J$
+are standard complex normal, this corresponds to 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
this spectrum using the saddle point of a replica symmetric calculation for the
Green function.
@@ -348,7 +350,7 @@ may be preformed for arbitrary $a$. For all values of $\kappa$ and $\epsilon$,
the resulting expression is always maximized for $a=\infty$. Taking this saddle
gives
\begin{equation} \label{eq:bezout}
- \ln \overline{\mathcal N}(\kappa,\epsilon)
+ \log\overline{\mathcal N}(\kappa,\epsilon)
={N\log(p-1)}
\end{equation}
This is precisely the Bézout bound, the maximum number of solutions that $N$
@@ -383,14 +385,14 @@ $a$-dependence gives a cumulative count, this implies a $\delta$-function
density of critical points along the line $y=0$. The number of critical points
contained within is
\begin{equation}
- \lim_{a\to1}\lim_{\kappa\to1}\overline{\mathcal N}(\kappa,0,a)
- \sim (p-1)^{N/2},
+ \lim_{a\to1}\lim_{\kappa\to1}\log\overline{\mathcal N}(\kappa,0,a)
+ = \frac12N\log(p-1),
\end{equation}
-the square root of \eqref{eq:bezout} and precisely the number of critical
+half of \eqref{eq:bezout} and corresponding precisely to the number of critical
points of the real pure spherical $p$-spin model. (note the role of conjugation
-symmetry, already underlined in Ref\cite{Bogomolny_1992_Distribution}). In
-fact, the full $\epsilon$-dependence of the real pure spherical $p$-spin is
-recovered by this limit as $\epsilon$ is varied.
+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}[htpb]
\centering