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authorJaron Kent-Dobias <jaron@kent-dobias.com>2020-12-10 14:50:58 +0100
committerJaron Kent-Dobias <jaron@kent-dobias.com>2020-12-10 14:50:58 +0100
commita1d8f3d7046eee32320ee35fde76e96777db06f6 (patch)
treed9dbea23515d0cce1e301ff5adf2ec3950997220
parentde26eed62ae3d28f1502c45e518f17c13740cf83 (diff)
parent8c8695de43dbc4d84c0441b2967b754d2a28dbda (diff)
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Merge branch 'master' of https://git.overleaf.com/5fcce4736e7f601ffb7e1484
-rw-r--r--bezout.bib7
-rw-r--r--bezout.tex27
2 files changed, 20 insertions, 14 deletions
diff --git a/bezout.bib b/bezout.bib
index c82da78..344196d 100644
--- a/bezout.bib
+++ b/bezout.bib
@@ -11,6 +11,13 @@
url = {https://doi.org/10.1007%2Fjhep12%282016%29071},
doi = {10.1007/jhep12(2016)071}
}
+@book{livan2018introduction,
+ title={Introduction to random matrices: theory and practice},
+ author={Livan, Giacomo and Novaes, Marcel and Vivo, Pierpaolo},
+ volume={26},
+ year={2018},
+ publisher={Springer}
+}
@article{Antenucci_2015_Complex,
author = {Antenucci, F. and Crisanti, A. and Leuzzi, L.},
diff --git a/bezout.tex b/bezout.tex
index c4560b5..b4b52e0 100644
--- a/bezout.tex
+++ b/bezout.tex
@@ -161,7 +161,7 @@ partner. For each pair $\pm\lambda$ of the real matrix, $\lambda^2$ 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 $\partial\partial H$, while both are the same as the
+singular values of $\partial\partial H$, the
distribution of square-rooted eigenvalues of $(\partial\partial
H)^\dagger\partial\partial H$.
@@ -172,7 +172,7 @@ study here, the {\em annealed approximation} $N \Sigma \sim \ln \overline{
\mathcal N_J} = \ln \int dJ \; N_J$ is exact.
A useful property of the Gaussian distributions is that gradient and Hessian
-may be seen to be independent \cite{Bray_2007_Statistics,
+for given $\epsilon$ may be seen to be independent \cite{Bray_2007_Statistics,
Fyodorov_2004_Complexity}, so that we may treat the $\delta$-functions and the
Hessians as independent. We compute each by taking the saddle point. The
$\delta$-functions are converted to exponentials by the introduction of
@@ -260,7 +260,7 @@ shift, or $\rho(\lambda)=\rho_0(\lambda+p\epsilon)$. The Hessian of
\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. Given its variances
+{\bf \color{red} restricting to directions proportional to $z$, i.e. orthogonal to the constraint}, these makes its ensemble that of Gaussian complex symmetric matrices. Given its variances
$\overline{|\partial_i\partial_j H_0|^2}=p(p-1)a^{p-2}/2N$ and
$\overline{(\partial_i\partial_j H_0)^2}=p(p-1)\kappa/2N$, $\rho_0(\lambda)$ is
constant inside the ellipse
@@ -274,17 +274,17 @@ spectrum of $\partial\partial H$ therefore is that of an ellipse in the complex
plane 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, or equivalently the
-eigenvalue spectrum of $(\partial\partial H)^\dagger\partial\partial H$, is the
+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 known. We have worked out an implicit form for
+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.
Introducing replicas to bring the partition function to
-the numerator of the Green function gives
+the numerator of the Green function \cite{livan2018introduction} gives
\begin{widetext}
\begin{equation} \label{eq:green.replicas}
G(\sigma)=\frac1N\lim_{n\to0}\int d\zeta\,d\zeta^*\,(\zeta_i^{(0)})^*\zeta_i^{(0)}
@@ -313,7 +313,7 @@ the numerator of the Green function gives
\end{widetext}
The argument of the exponential has several saddles. The solutions $\alpha_0$
are the roots of a sixth-order polynomial, but the root with the
-smallest value of $\mathop{\mathrm{Re}}\alpha_0$ appears gives the correct
+smallest value of $\mathop{\mathrm{Re}}\alpha_0$ in all the cases we studied gives the correct
solution. A detailed analysis of the saddle point integration is needed to
understand why this is so. Given such $\alpha_0$, the density of singular
values follows from the jump across the cut, or
@@ -345,19 +345,18 @@ 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}
- \overline{\mathcal N}(\kappa,\epsilon)
- =e^{N\log(p-1)}
- =(p-1)^N.
+ \ln \overline{\mathcal N}(\kappa,\epsilon)
+ ={N\log(p-1)}
\end{equation}
This is precisely the Bézout bound, the maximum number of solutions that $N$
equations of degree $p-1$ may have \cite{Bezout_1779_Theorie}. More insight is
gained by looking at the count as a function of $a$, defined by
\begin{equation} \label{eq:count.def.marginal}
- \overline{\mathcal N}(\kappa,\epsilon)
- =\int da\,\overline{\mathcal N}(\kappa,\epsilon,a)
+ {\mathcal N}(\kappa,\epsilon,a)
+ ={\mathcal N}(\kappa,\epsilon/ \sum_i y_i^2<Na)
\end{equation}
and likewise the $a$-dependant complexity
-$\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal N}(\kappa,\epsilon,a)$. In
+$\Sigma(\kappa,\epsilon,a)=N\log\overline{\mathcal N}_a(\kappa,\epsilon,a)$
the large-$N$ limit, the $a$-dependant expression may be considered the
cumulative number of critical points up to the value $a$.