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Diffstat (limited to 'marginal.tex')
| -rw-r--r-- | marginal.tex | 22 |
1 files changed, 18 insertions, 4 deletions
diff --git a/marginal.tex b/marginal.tex index c461f81..e2bd24f 100644 --- a/marginal.tex +++ b/marginal.tex @@ -288,13 +288,27 @@ corresponds to the intersection of the average spectrum with zero, i.e., a pseudogap. \begin{figure} - \includegraphics[width=\columnwidth]{figs/large_deviation.pdf} + \hspace{1.3em} + \includegraphics{figs/spectrum_less.pdf} + \hspace{-2em} + \includegraphics{figs/spectrum_eq.pdf} + \hspace{-2em} + \includegraphics{figs/spectrum_more.pdf} + \\ + \includegraphics{figs/large_deviation.pdf} \caption{ - The large deviation function $G_\sigma(\mu)$ defined in + The large deviation function $G_0(\mu)$ defined in \eqref{eq:large.dev} as a function of the shift $\mu$ to the - GOE diagonal. As expected, $G_\sigma(2\sigma)=0$, while for + GOE diagonal. $G_0(2\sigma)=0$, while for $\mu>2\sigma$ it is negative and for $\mu<2\sigma$ it gains an - imaginary part. + imaginary part. The top panels show schematically what happens to the + spectral density in each of these regimes. For $\mu<2\sigma$, an $N^2$ + large deviation would be required to fix the smallest eigenvalue to zero + and the calculation breaks down, leading to the imaginary part. For + $\mu>2\sigma$ the spectrum can satisfy the constraint on the smallest + eigenvalue by isolating a single eigenvalue at zero at the cost of an + order-$N$ large deviation. At $\mu=2\sigma$, the transition point, the + spectrum is pseudogapped or marginal. } \label{fig:large.dev} \end{figure} |