Spectral measure of GOE matrices
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I am studying a paper and there is a paragraph which describes some notation on random matrices.. I quote :
" The GOE ensemble is a probability measure on the space of real symmetric $Ntimes N$matrices. Namely, it is the prob distribution of the $N times N$ real symmetric matrices $M$ whose entries are independent $N(0,1)$ random variables with variance $E[M_{ij}^2] = frac{1+ delta _{ij}} {2N} $.
We denote by $E_{GOE}$ the expectation under the GOE ensemble.
Let $lambda_0^N leqslant lambda_1^N leqslant ... leqslant lambda_{N-1}^N $ be the ordered eigenvalues of $M$.
We denote by $L_N = frac{1}{N} sum limits_{i} delta_{lambda_{i}^{N}}$ the (random) spectral measure of $M$, and by $rho_N (x)$ the density of the (non-random) probability measure $E_{GOE} (L_N)$ which satisfies $int_mathbb{R} f(x)rho_N(x)dx = frac{1}{N} E_{GOE}[sumlimits_{i=0}^{N-1} f(lambda_{i}^{N})] $. "
I have two questions : 1)what are the maps of $lambda_{i}^{N}$ and of $L_N$ (ie arguments and values)
and 2) why is $E_{GOE}(L_N)$ a probability measure on $mathbb{R}$ ?
Could someone give me some help ?
random-matrices
asked Dec 27 '18 at 20:06
vl.athvl.ath
749
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add a comment |
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I am studying a paper and there is a paragraph which describes some notation on random matrices.. I quote :
" The GOE ensemble is a probability measure on the space of real symmetric $Ntimes N$matrices. Namely, it is the prob distribution of the $N times N$ real symmetric matrices $M$ whose entries are independent $N(0,1)$ random variables with variance $E[M_{ij}^2] = frac{1+ delta _{ij}} {2N} $.
We denote by $E_{GOE}$ the expectation under the GOE ensemble.
Let $lambda_0^N leqslant lambda_1^N leqslant ... leqslant lambda_{N-1}^N $ be the ordered eigenvalues of $M$.
We denote by $L_N = frac{1}{N} sum limits_{i} delta_{lambda_{i}^{N}}$ the (random) spectral measure of $M$, and by $rho_N (x)$ the density of the (non-random) probability measure $E_{GOE} (L_N)$ which satisfies $int_mathbb{R} f(x)rho_N(x)dx = frac{1}{N} E_{GOE}[sumlimits_{i=0}^{N-1} f(lambda_{i}^{N})] $. "
I have two questions : 1)what are the maps of $lambda_{i}^{N}$ and of $L_N$ (ie arguments and values)
and 2) why is $E_{GOE}(L_N)$ a probability measure on $mathbb{R}$ ?
Could someone give me some help ?
random-matrices
asked Dec 27 '18 at 20:06
vl.athvl.ath
749
$endgroup$
add a comment |
$begingroup$
I am studying a paper and there is a paragraph which describes some notation on random matrices.. I quote :
" The GOE ensemble is a probability measure on the space of real symmetric $Ntimes N$matrices. Namely, it is the prob distribution of the $N times N$ real symmetric matrices $M$ whose entries are independent $N(0,1)$ random variables with variance $E[M_{ij}^2] = frac{1+ delta _{ij}} {2N} $.
We denote by $E_{GOE}$ the expectation under the GOE ensemble.
Let $lambda_0^N leqslant lambda_1^N leqslant ... leqslant lambda_{N-1}^N $ be the ordered eigenvalues of $M$.
We denote by $L_N = frac{1}{N} sum limits_{i} delta_{lambda_{i}^{N}}$ the (random) spectral measure of $M$, and by $rho_N (x)$ the density of the (non-random) probability measure $E_{GOE} (L_N)$ which satisfies $int_mathbb{R} f(x)rho_N(x)dx = frac{1}{N} E_{GOE}[sumlimits_{i=0}^{N-1} f(lambda_{i}^{N})] $. "
I have two questions : 1)what are the maps of $lambda_{i}^{N}$ and of $L_N$ (ie arguments and values)
and 2) why is $E_{GOE}(L_N)$ a probability measure on $mathbb{R}$ ?
Could someone give me some help ?
random-matrices
asked Dec 27 '18 at 20:06
vl.athvl.ath
749
$endgroup$
I am studying a paper and there is a paragraph which describes some notation on random matrices.. I quote :
" The GOE ensemble is a probability measure on the space of real symmetric $Ntimes N$matrices. Namely, it is the prob distribution of the $N times N$ real symmetric matrices $M$ whose entries are independent $N(0,1)$ random variables with variance $E[M_{ij}^2] = frac{1+ delta _{ij}} {2N} $.
We denote by $E_{GOE}$ the expectation under the GOE ensemble.
Let $lambda_0^N leqslant lambda_1^N leqslant ... leqslant lambda_{N-1}^N $ be the ordered eigenvalues of $M$.
We denote by $L_N = frac{1}{N} sum limits_{i} delta_{lambda_{i}^{N}}$ the (random) spectral measure of $M$, and by $rho_N (x)$ the density of the (non-random) probability measure $E_{GOE} (L_N)$ which satisfies $int_mathbb{R} f(x)rho_N(x)dx = frac{1}{N} E_{GOE}[sumlimits_{i=0}^{N-1} f(lambda_{i}^{N})] $. "
I have two questions : 1)what are the maps of $lambda_{i}^{N}$ and of $L_N$ (ie arguments and values)
and 2) why is $E_{GOE}(L_N)$ a probability measure on $mathbb{R}$ ?
Could someone give me some help ?
random-matrices
random-matrices
asked Dec 27 '18 at 20:06
vl.athvl.ath
749
asked Dec 27 '18 at 20:06
vl.athvl.ath
749
asked Dec 27 '18 at 20:06
vl.athvl.ath
749
asked Dec 27 '18 at 20:06
vl.athvl.ath
749
asked Dec 27 '18 at 20:06
vl.athvl.ath
749
749
add a comment |
add a comment |
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