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(*deployswithcanonicalname*)deploy:=Module[{notebookFn,parentDir,cloudFn,result},Print[DateString[]];notebookFn=FileNameSplit[NotebookFileName[]][[-1]];parentDir=FileNameSplit[NotebookFileName[]][[-2]];cloudFn=parentDir~StringJoin~"/"~StringJoin~notebookFn;result=CloudDeploy[SelectedNotebook[],CloudObject[cloudFn],Permissions"Public",SourceLinkNone];Print["Uploading to ",cloudFn];result];deploy
Thu 2 Mar 2023 16:47:12
Eigenvalue decay for semi-circle law
Eigenvalue decay for semi-circle law
Background on semi-circle law https://math.stackexchange.com/a/4187243/998
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d=1000;eigs=Reverse@Sort@Abs@Eigenvalues@N@AdjacencyMatrix@RandomGraph@BernoulliGraphDistribution[d,0.5];eigs=Rest[eigs];(*Firsteigenvalueisoutlier*)eigs=eigs/First[eigs];(*Normalizetohavefirsteigenvalueat1*)xvals=Range[0,1-2/d,1/d];(*remapfrom0..dtoto0..1range*)observed=ListPlot[{xvals,eigs},PlotStyle->Directive[Opacity[.2],Red],PlotLegends->{"observed"}];Assuming[{0<y<1},icdf[y_]=InverseCDF[WignerSemicircleDistribution[1],y]//Refine];g[x_]=icdf[(1-x/2)];(*reversesort,map0.5->1rangeto0..1*)Print["Predicted law: ",g[x_]];predicted=Plot[g[x],{x,0,1},FillingAxis,PlotRangeAll,PlotStyle->Black,PlotLegends{"predicted"}];Show[observed,predicted]harmonic=1/(2x+1);LogPlot[{g[x],harmonic},{x,0,.9},FillingAxis,PlotRangeAll,PlotLegends->{"semicircle",harmonic},PlotLabel->"Harmonic approximation of spectrum"]
Predicted law: -1+2InverseBetaRegularized1-,,
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