Spectrum of product of random matrices

Related:

-- mathematica.SE post

In[]:=

ClearAll["Global`*"];SeedRandom[1];spectralNormalize[evals_]:=evals/Max[evals];SF=StringForm;If[SUPERLARGE`SUPERLARGE===1,n=1000;numSamples=100;Clear[SUPERLARGE`SUPERLARGE];x0=

-20

10

;Print["Running large"],n=100;numSamples=10;x0=

-12

10

;];(*SingleGaussianmatrix*)sample:=RandomVariate[NormalDistribution[],{n,n}];(*Productofkrandommatrices*)randomProduct[k_]:=Nest[#.sample&,sample,k-1];(*CDFoflistofnumbers*)genCDF[coefs_]:={Sort@coefs,Range@Length@coefs/Length[coefs]};sf=None;sf={"Log","Log"};kvals=Range[5];legends=SF["k=``",#]&/@kvals;guesses=

1/2

x

,

1/3

x

,

1/4

x

,

1/5

x

;(*Productofkrandommatrices*)productEvals[k_]:=

2

(spectralNormalize@SingularValueList[randomProduct@k])

;(*Computeseveralproducts,pooltheirspectra*)batchedProductEvals[k_,samples_]:=Flatten@Table[productEvals[k],samples];observedPlot=ListLinePlot[Table[genCDF[batchedProductEvals[k,numSamples]],{k,kvals}],ScalingFunctions->sf,PlotLegends->legends];predictedPlot=Plot@@{guesses,{x,x0,1},ScalingFunctions->sf,PlotStyle->Dashed,PlotLegends->guesses};Show[observedPlot,predictedPlot,PlotLabel->SF["CDF of σ(FractionBox[\(A\), \(\(||\)\(A\)\(||\)\)]

2

)\),

for A=product of k IID matrices",n,n],AxesLabel->{"

2

σ

"}]

Running large

Out[]=

	k=1
	k=2
	k=3
	k=4
	k=5

	x
	1/3 x
	1/4 x
	1/5 x

In[]:=

SUPERLARGE`SUPERLARGE=1

Out[]=

1