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| -rw-r--r-- | bezout.tex | 16 |
1 files changed, 9 insertions, 7 deletions
@@ -30,7 +30,7 @@ solutions averaged over randomness in the $N\to\infty$ limit. We find that it saturates the Bézout bound $\log\overline{\mathcal{N}}\sim N \log(p-1)$. The Hessian of each saddle is given by a random matrix of the form $C^\dagger - C$, where $C$ is a complex Gaussian matrix with a shift to its diagonal. Its + C$, where $C$ is a complex {\color{red} symmetric} Gaussian matrix with a shift to its diagonal. Its spectrum has a transition where a gap develops that generalizes the notion of `threshold level' well-known in the real problem. The results from the real problem are recovered in the limit of real parameters. In this case, only the @@ -95,7 +95,7 @@ introducing $\epsilon\in\mathbb C$, our energy is \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. We choose to + We choose to constrain our model by $z^2=N$ rather than $|z|^2=N$ in order to preserve the analyticity of $H$. The nonholomorphic constraint also has a disturbing lack of critical points nearly everywhere: if $H$ were so constrained, then @@ -104,22 +104,24 @@ $0=\partial^* H=-p\epsilon z$ would only be satisfied for $\epsilon=0$. The critical points are of $H$ given by the solutions to the set of equations \begin{equation} \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}} - = p\epsilon z_i + = p\epsilon z_i \label{cosa} \end{equation} for all $i=\{1,\ldots,N\}$, which for fixed $\epsilon$ is a set of $N$ -equations of degree $p-1$, to which one must add the constraint. In this sense +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 $N=1$ and $p\to\infty$. +We see from (\ref{cosa}) that at any critical point, $\epsilon=H/N$, the average energy. Since $H$ is holomorphic, any critical point of $\operatorname{Re}H$ is also a critical point of $\operatorname{Im}H$. The number of critical points of $H$ is -therefore the same as that of $\operatorname{Re}H$. From each critical point +therefore the same as that of $\operatorname{Re}H$. From each saddle emerges a gradient line of $\operatorname{Re}H$, which is also one of constant $\operatorname{Im}H$ and therefore constant phase. Writing $z=x+iy$, $\operatorname{Re}H$ can be considered a real-valued function -of $2N$ real variables. Its number of critical points is given by the usual +of $2N$ real variables. Its number of saddle-points is given by the usual Kac--Rice formula: \begin{equation} \label{eq:real.kac-rice} \begin{aligned} @@ -152,7 +154,7 @@ transformations through, we have \end{equation} This gives three equivalent expressions for the determinant of the Hessian: as that of a $2N\times 2N$ real matrix, that of an $N\times N$ Hermitian matrix, -or the norm squared of that of an $N\times N$ complex symmetric matrix. +i.e. the norm squared of that of an $N\times N$ complex symmetric matrix. These equivalences belie a deeper connection between the spectra of the corresponding matrices. Each positive eigenvalue of the real matrix has a |