From 6df768b8aa400f6e0ab2017797c928dcc831cd64 Mon Sep 17 00:00:00 2001 From: Jaron Kent-Dobias Date: Fri, 7 Jun 2024 17:18:14 +0200 Subject: Modified large deviation figure. --- figs/large_deviation.pdf | Bin 11693 -> 11522 bytes figs/spectrum_eq.pdf | Bin 0 -> 7907 bytes figs/spectrum_less.pdf | Bin 0 -> 11112 bytes figs/spectrum_more.pdf | Bin 0 -> 8089 bytes figures.nb | 3240 ++++++++++++++++++++++++---------------------- marginal.tex | 22 +- 6 files changed, 1681 insertions(+), 1581 deletions(-) create mode 100644 figs/spectrum_eq.pdf create mode 100644 figs/spectrum_less.pdf create mode 100644 figs/spectrum_more.pdf diff --git a/figs/large_deviation.pdf b/figs/large_deviation.pdf index 9149a5a..29424c2 100644 Binary files a/figs/large_deviation.pdf and b/figs/large_deviation.pdf differ diff --git a/figs/spectrum_eq.pdf b/figs/spectrum_eq.pdf new file mode 100644 index 0000000..69db15c Binary files /dev/null and b/figs/spectrum_eq.pdf differ diff --git a/figs/spectrum_less.pdf b/figs/spectrum_less.pdf new file mode 100644 index 0000000..439d485 Binary files /dev/null and b/figs/spectrum_less.pdf differ diff --git a/figs/spectrum_more.pdf b/figs/spectrum_more.pdf new file mode 100644 index 0000000..8db227a Binary files /dev/null and b/figs/spectrum_more.pdf differ diff --git a/figures.nb b/figures.nb index 411cf9a..3a5b708 100644 --- a/figures.nb +++ b/figures.nb @@ -10,10 +10,10 @@ NotebookFileLineBreakTest NotebookFileLineBreakTest NotebookDataPosition[ 158, 7] -NotebookDataLength[ 1187648, 23952] -NotebookOptionsPosition[ 1168333, 23646] -NotebookOutlinePosition[ 1168732, 23662] -CellTagsIndexPosition[ 1168689, 23659] +NotebookDataLength[ 1191909, 24038] +NotebookOptionsPosition[ 1172284, 23727] +NotebookOutlinePosition[ 1172683, 23743] +CellTagsIndexPosition[ 1172640, 23740] WindowFrame->Normal*) (* Beginning of Notebook Content *) @@ -25,18 +25,25 @@ Cell["Settings", "Section", 3.915530723762684*^9}},ExpressionUUID->"926df485-ea0b-4c71-a1d6-\ 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-> <||>, - "PlotRange" -> {{-15., 3.}, {0., 0.12732395447351627`}}, - "Frame" -> {{False, False}, {False, False}}, - "AxesOrigin" -> {0, 0}, "ImageSize" -> {180, 72}, - "Axes" -> {True, True}, "LabelStyle" -> {FontFamily -> "Helvetica", - GrayLevel[0], FontSize -> 12}, "AspectRatio" -> Rational[2, 5], + "PlotRange" -> {{-3., 15.}, {0., 0.16976527263135505`}}, + "Frame" -> {{True, True}, {True, True}}, "AxesOrigin" -> {0, 0}, + "ImageSize" -> {118, + Rational[236, 3]}, "Axes" -> {True, True}, + "LabelStyle" -> {FontFamily -> "Times", + GrayLevel[0], FontSize -> 11}, "AspectRatio" -> Rational[2, 3], "DefaultStyle" -> { Directive[ Opacity[1.], @@ -1451,11 +1511,12 @@ Jzo/BszwObXG19vhfxtewnk= "Label" -> {"XYLabel"}, "Ball" -> {"InterpolatedBall"}|>, "LayoutOptions" -> <| "PanelPlotLayout" -> <||>, - "PlotRange" -> {{-15., 3.}, {0., 0.12732395447351627`}}, - "Frame" -> {{False, False}, {False, False}}, "AxesOrigin" -> {0, 0}, - "ImageSize" -> {180, 72}, "Axes" -> {True, True}, - "LabelStyle" -> {FontFamily -> "Helvetica", - GrayLevel[0], FontSize -> 12}, "AspectRatio" -> Rational[2, 5], + "PlotRange" -> {{-3., 15.}, {0., 0.16976527263135505`}}, + "Frame" -> {{True, True}, {True, True}}, "AxesOrigin" -> {0, 0}, + "ImageSize" -> {118, + Rational[236, 3]}, "Axes" -> {True, True}, + "LabelStyle" -> {FontFamily -> "Times", + GrayLevel[0], FontSize -> 11}, "AspectRatio" -> Rational[2, 3], "DefaultStyle" -> { Directive[ Opacity[1.], @@ -1481,146 +1542,145 @@ Jzo/BszwObXG19vhfxtewnk= RGBColor[0.368417, 0.506779, 0.709798], AbsoluteThickness[2]], Line[CompressedData[" -1:eJwV13k4lG8XB3BrsoRmsy8xQxsKKW0n2ohKpUhKiRChlBZly1LKFtpkiUL2 -ZJl5RuYoREQ/2SprImTJEi3ifd6/5vpczzXXnJn73N9zZom9x35HAT4+Pl9+ -Pr7/v05+N9WemGfhhPfWTRcebQOfRusDG/+xkLv8aUK9+zloOpU4eeIPC18V -MWvUt18Brb+9MaEzLFxS3PBIQ9UfQiKX6edMsdBjZ3RNf2ogdKl7NDWOs7BB -YYUw/VMQrGMXnv81ykIFjUPfjT+GgPGQodfKYRY6+oZPfXt6A8wVeWePD7Iw -4jwvt3x/GNj517hX97JQq1L0n5d9ODgX7Dkz283CrIvW/hl1EXDu6wfXVZ0s 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-> "Times", + GrayLevel[0], FontSize -> 11}, "AspectRatio" -> Rational[2, 3], "DefaultStyle" -> { Directive[ Opacity[1.], @@ -2636,11 +2712,12 @@ X0RF02h4sPWPSYyNDfwfBVQvfw== "Label" -> {"XYLabel"}, "Ball" -> {"InterpolatedBall"}|>, "LayoutOptions" -> <| "PanelPlotLayout" -> <||>, - "PlotRange" -> {{-15., 3.}, {0., 0.12732395447351627`}}, - "Frame" -> {{False, False}, {False, False}}, "AxesOrigin" -> {0, 0}, - "ImageSize" -> {180, 72}, "Axes" -> {True, True}, - "LabelStyle" -> {FontFamily -> "Helvetica", - GrayLevel[0], FontSize -> 12}, "AspectRatio" -> Rational[2, 5], + "PlotRange" -> {{-3., 15.}, {0., 0.16976527263135505`}}, + "Frame" -> {{True, True}, {True, True}}, "AxesOrigin" -> {0, 0}, + "ImageSize" -> {118, + Rational[236, 3]}, "Axes" -> {True, True}, + "LabelStyle" -> {FontFamily -> "Times", + GrayLevel[0], FontSize -> 11}, "AspectRatio" -> Rational[2, 3], "DefaultStyle" -> { Directive[ Opacity[1.], @@ -2666,143 +2743,144 @@ X0RF02h4sPWPSYyNDfwfBVQvfw== RGBColor[0.368417, 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\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} -- cgit v1.2.3-70-g09d2